Preprint typeset in JHEP style - HYPER VERSION IFT{UAM ... · Preprint typeset in JHEP style - HYPER VERSION IFT{UAM/CSIC{09 {43 MPP{2009{162 PUPT-2315 Quasinormal modes of massive

Preprint typeset in JHEP style - HYPER VERSION IFT–UAM/CSIC–09–43MPP–2009–162

PUPT-2315

Quasinormal modes of massive charged flavor branes

Matthias Kaminskia,b, Karl Landsteinera and Francisco Pena-Beniteza

a Instituto de Fısica Teorica CSIC/UAM, C-XVI Universidad Autonoma de MadridE-28049 Madrid, Spain

b Department of Physics, Princeton University, Princeton, NJ 08544, USA.E-mail: [email protected], karl.landsteiner, [email protected]

Johanna Erdmenger, Constantin Greubel, Patrick Kerner

Max-Planck-Institut fur Physik (Werner-Heisenberg-Institut)Fohringer Ring 6, 80805 Munchen, GermanyE-mail: jke, greubel, [email protected]

Abstract: We present an analysis and classification of vector and scalar fluctuations ina D3/D7 brane setup at finite termperature and baryon density. The system is dual toan N = 2 supersymmetric Yang-Mills theory with SU(Nc) gauge group and Nf hyper-multiplets in the fundamental representation in the quenched approximation. We improvesignificantly over previous results on the quasinormal mode spectrum of D7 branes andstress their novel physical interpretation. Amongst our findings is a new purely imaginaryscalar mode that becomes tachyonic at sufficiently low temperature and baryon density.We establish the existence of a critical density above which the scalar mode stays in thestable regime for all temperatures. In the vector sector we study the crossover from thehydrodynamic to the quasiparticle regime and find that it moves to shorter wavelengths forlower temperatures. At zero baryon density the quasinormal modes move toward distinctdiscrete attractor frequencies that depend on the momentum as we increase the temper-ature. At finite baryon density, however, the trajectories show a turning behavior suchthat for low temperature the quasinormal mode spectrum approaches the spectrum of thesupersymmetric zero temperature normal modes. We interpret this as resolution of thesingular quasinormal mode spectrum that appears at the limiting D7 brane embedding atvanishing baryon density.

Keywords: Gauge-gravity correspondence, D-branes, Black Holes.

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Contents

1. Introduction and summary 2

2. Quasinormal modes and Holography 6

3. Holographic Setup 73.1 The ρ-coordinate system 83.2 Brane setup and background fields in z-coordinates 9

4. Vanishing momentum and density 114.1 Transverse vectors 114.2 Longitudinal vectors 124.3 Scalar 144.4 Schrodinger potential analysis 154.5 Discussion: Tachyon and De-singularization 16

5. Finite momentum but vanishing density 175.1 Transverse vectors 175.2 Longitudinal vectors 195.3 Scalar 195.4 Schrodinger potential analysis 215.5 Discussion: Breakdown of hydrodynamics, ’attractors’ and the tachyon 23

5.5.1 Hydrodynamics to Collisionless Crossover 235.5.2 ”Attractor” frequencies 255.5.3 Tachyon: A new hydrodynamic mode 26

6. Finite density but vanishing momentum 276.1 Transverse vectors 276.2 Scalar 286.3 Schrodinger potential analysis 316.4 Analytic solution at high frequencies 356.5 Discussion: Turning point and Tachyon 38

6.5.1 Turning point 396.5.2 Killing the Tachyon 43

7. Conclusions and Outlook 43

A. Shooting method 45

B. Relaxation method 46

C. Schrodinger potentials 48

– 1 –

D. Finite density but vanishing momentum 50

E. Results for second quasinormal modes 51E.1 Transverse vector fluctuations 51E.2 Longitudinal vector fluctuations 52E.3 Scalar fluctuations 53

1. Introduction and summary

One of the successes of the AdS/CFT correspondence [1, 2] and its generalizations is itsapplication to the plasma phase of non-Abelian gauge theories [3]. This is of high interestbecause of its potential relevance for the description of the strongly-coupled quark-gluonplasma as it is created in Heavy Ion Collision at RHIC or in the near future at the LHC.

The holographic modelling of the plasma phase invariably involves an asymptoticallyAdS black hole [4]. Of particular interest are the quasinormal modes of such black holesas they are mapped to the poles in the correlation function of the dual finite temperaturefield theory [5,6]. One aspect of this is a relation between the quasinormal frequencies andthe hydrodynamic transport coefficients, e. g. the shear viscosity.

An important generalization of the AdS/CFT correspondence is the addition of flavordegrees of freedom in the fundamental representation of the gauge group. One convenientway of achieving this is via the addition of D7-brane probes to the ten-dimensional super-gravity background [7]. The meson spectrum can then be studied via the fluctuations ofthe probe brane [8, 9].

For the configuration of a D7-brane probe added to the AdS Schwarzschild black holebackground, a first order phase transition occurs between D7-brane probes either stayingoutside of the horizon or reaching down to it [10–12]. These two types of embeddings arecalled Minkowski or black hole embeddings, respectively. As discussed in [13], the firstcase corresponds to stable mesons, whereas in the second case the mesons are unstable.For stable mesons, the spectrum has been determined analytically at zero temperaturein [8]. In the second case, the meson excitations may be identified with quasinormal modesand their finite width is related to an infalling energy boundary condition at the blackhole horizon. The spectral functions for this configuration were first studied in [14]. If thetemperature is high compared to the quark mass the quasinormal frequencies lie deep insidethe lower complex half plane and the spectral function is smooth and without noticeablepeaks. As the temperature is lowered the quasinormal modes move towards the real axis,producing rather well defined quasiparticle peaks in the spectral function. This behavioris particularly strong at finite baryon density.

Furthermore a tachyonic mode is present in the scalar sector close to the first or-der phase transition discussed above. For null momentum, the spectral function of this

– 2 –

configuration has been studied in [15] and the presence of the tachyonic mode has beenargued for by way of a WKB analysis. We find indeed a so far overlooked purely imaginarymode in the scalar spectrum that crosses the real axis and therefore becomes unstable ata temperature that is in excellent numerical agreement with [15].

At finite baryon chemical potential, as obtained by considering a non-zero profile forthe time component of the gauge field on the D7-brane, there is a rich phase structure forthe D7-brane embedded in the AdS Schwarzschild background [16–20]. In these works thestrict supergravity limit is considered, i.e. α′ → 0, and the results are thus perturbative onthe gravity side. Non-perturbative effects such as worldsheet instantons have been studiedin [21]. Taking these effects into account the first-order phase transition mentioned abovebecomes third order at a critical chemical potential [21]. In this paper however we willconsider the strict supergravity limit only.

The spectral functions for vector mesons at finite baryon density have been studiedin [22], where a particular turning behavior of the quasi-normal frequencies in the complexfrequency plane was observed as function of the ratio of quark mass over temperature: Atlow values of Mq/T , the position of the quasinormal modes moves to smaller real partswhen Mq/T increases, while for large values, it moves to larger real parts. For large Mq/T ,the spectrum approaches the form of the supersymmetric case. The momentum dependenceof the spectral function at finite baryon density has been investigated in [23,24].

Summary of results The purpose of the present paper is to present an in depth studyof both scalar and vector modes at either finite baryon density or finite momentum.

At zero density and zero momentum we determine the first and second quasinormalfrequency of the scalar and vector fluctuations. We calculate their trajectories in thecomplex plane as we increase the quark mass over temperature ratio. In general thesemodes move along a curve where initially the real and imaginary parts can grow a littlebut then move continuously towards lower values. As shown in [25] the endpoints ofthese curves asymptote to a single point on the real frequency axis that is reached as theembedding reaches the limiting embedding. This happens when the brane just touches theblack hole horizon1. There it has been argued that the spectrum of quasinormal modesbecomes singular as they all coincide at one (real) energy value. It is however unclear if thislimiting embedding can really be reached since the embeddings become locally unstableat rather high temperatures. Indeed in the scalar sector we find a mode that becomestachyonic shortly after the system has become thermodynamically metastable due to thepresence of the first order phase transition. Once the tachyonic mode is in the spectrumthe embedding is not even metastable and so far it is not known what the true groundstate would be or if such a state exists at all.

At finite momentum but zero density, we find that at a critical momentum the systemundergoes a crossover transition from the hydrodynamic behavior at long wavelengths to

1Note that this limiting embedding is often called critical embedding in the literature e. g. [12]. However

in this paper we use the term ‘critical embedding’ for the embedding at which the phase transition occurs

(cf. figure 1 a).

– 3 –

a collisionless behavior at small wavelengths. We observe this crossover in the longitudinalvector channel which has a hydrodynamic mode, i. e. a mode whose dispersion relationdoes not show a gap at zero momentum. This mode describes baryon charge diffusion andis thus called diffusion mode. The transition from the hydrodynamic to the collisionlessregime is defined as the point where the imaginary part of the diffusion mode becomeslarger than the imaginary part of the first quasinormal mode which is not a hydrodynamicmode. Additionally we observe that the first quasinormal mode of the scalar and vectorfluctuations with increasing momentum show an increasing number of spirals in their tra-jectories parametrized by the quark mass over temperature ratio. The number n of thesespirals apparently is correlated with certain ”attractor” frequencies, i.e. the real part ofthe quasinormal modes asymptotes to ωn ∈ R as the D7-brane embedding approachesthe critical embedding between the black hole and Minkowski embeddings. Although theasymptotic values ωn lie deep in the unstable sector, the spirals mostly lie in the physicalsector of the theory. So they are physical features of the (meta)stable theory.

At finite quark density but zero momentum, we find the following: In the vector modesector, there are two distinct movements of the quasinormal frequencies in dependenceof the quark mass over temperature ratio. At small densities there is a turning point asalready observed in [22] and as described above. This turning point is no longer presentfor large densities. The critical value for the normalized density is dc = 0.04. The relationbetween the normalized density d and the baryon density nB is given by

d =2

52nB

Nf

√λT 3

, (1.1)

where T is the temperature, Nf the number of flavors and λ the ’t Hooft coupling. In thescalar channel, the quasinormal frequencies show also distinct movements in dependenceof the quark mass over temperature ratio. In this case we find three distinct behaviorsseparated by critical densities. As a function of Mq/T , the quasinormal modes performeither a left turn, a right turn or no turn in the complex frequency plane. We will studythese different behaviors in detail.

For the scalar channel, we find in addition that the tachyonic mode mentioned aboveis only present below a critical normalized density, d < 0.00315. We identify the unstableregion in which the tachyon is present with an instability related to the first order phasetransition found in the thermodynamics of the system. This behavior may be understoodby comparing it to the first order phase transition of the well-known van-der-Waals gas, asdescribed for instance in [26]. Figure 1 (b) shows the first order phase transition of the van-der-Waals gas, which follows the red line in the p-V -diagram as obtained from the Maxwellconstruction. Additionally there are two metastable branches: One of them reaches fromthe point where the phase transition begins when increasing the volume to the minimumdenoted by the red plus sign in the figure. The second reaches from the maximum denotedby the red cross to the point where the phase transition ends when increasing the volume.The region between the two extrema is unstable since the pressure raises when the volumeis increased.

– 4 –

Our calculations show a very similar behavior of the system considered here. We havecalculated both the free energy as function of m ∝ Mq/T close to the phase transition,2 displayed schematically in Figure 1 (a), and the quasinormal modes for the same rangeof Mq/T . Both calculations show a similar structure of stable, metastable and unstablebranches, with the numerical values for the boundaries of these branches agreeing up tofour digits in both calculations. In particular in analogy to the van-der-Waals gas we findthat in addition to the stable branches3 there are metastable branches close to the phasetransition, denoted as overheated and undercooled4 in figure 1 (a). Furthermore we findan unstable branch which connects these two metastable branches. The metastable phasesare stable against fluctuations, while on the unstable branch a tachyonic mode appears inthe quasinormal spectrum. In particular we emphasize that at zero density the limitingembedding which has a conical singularity at the black hole horizon, and is often discussedin the literature, lies clearly deep inside the unstable region. Thus any observation foundby using this embedding should be considered with great care since it does not correspondto a physical state. Furthermore we should stress that on the field theory side there existsa stable ground state for any combination of the mass m and chemical potential µ.

unstable

overheated

undercooled

phase transition

stable

m

F

(a)

PSfrag repla ements unstable bran hmetastable bran hes

stable bran hesphase transition

VP(V)=RT Vba V2

(b)

Figure 1: (a) Sketch of the free energy F of the flavor fields versus the quark mass over temperatureratio m ∝Mq/T close to the first order phase transition. (b) Pressure versus volume of the van-der-Waals gas. The red line marks the phase transition which is obtained by the Maxwell construction.

We also present a qualitative analysis of the quasinormal spectrum at either finitemomentum or finite density which uses the fact that the equation of motion for the fluctu-ations can be transformed into a Schrodinger equation. The Schrodinger potential analysiswas introduced for instance in [13,14,23,25].

2A similar calculation of the free energy was performed in [16].3Note that in [16] it was observed that for a small region of the stable phase in which the mass parameter

m is just slightly larger than the critical one there is an charge instability. This instability is distinct from

the instability discussed above. We do not observe this charge instability in our quasinormal mode analysis.

This may be due to inadequacy of the our numerical methods at large mass over temperature ratios.4In the context of the D3/D7 system, the terms ‘overheated’ and ‘undercooled’ were introduced in [25].

– 5 –

The results of this analysis are in qualitative agreement with the results obtained bystating the quasinormal modes listed above: In particular, at finite density we find a generalfeature in the Schrodinger potential: At low quark mass over temperature ratio there areonly unbound scattering states which correspond to quasinormal modes. By increasingthe quark mass over temperature ratio, a barrier forms in the Schrodinger potential [23]which leads to a local minimum. In [25] it was found for Minkowski embeddings that theSchrodinger potential is a box whose extent in the AdS radial direction coincides with theregion filled by the D7-brane probe. Here we observe that for black hole embeddings atfinite density, a barrier forms in the Schrodinger potential at the same radial position as theIR boundary of the box in the Minkowski case. This barrier separates the bulk of the AdSspace from the horizon of the black hole. As the barrier increases when the ratio of quarkmass over temperature m ∝ Mq/T raises, less energy can leak into the black hole. Thus‘bound’ states which correspond to ‘normal’ modes are formed. We study the appearanceand the behavior of this barrier in detail.

The paper is organized as follows: In section 2 we give a general introduction tothe relation between quasinormal modes and hydrodynamics. In section 3 we introducethe D3/D7-brane setup. In the sections 4, 5 and 6 we determine the quasinormal modesof the scalar and vector fluctuations first at vanishing momentum and density and laterat finite momentum or finite density. In every section we also calculate the correspondingSchrodinger potentials which we use to describe the qualitative behavior of the quasinormalmodes. At the end of each section we summarize the physical results which we found.

2. Quasinormal modes and Holography

In this section we recall the definition of quasinormal modes of black holes and the rolethey play in determining the response of a holographic field theory close to equilibrium.

Quasinormal modes of a black hole are distinct perturbations of the black hole solution.Roughly they can be understood as resonances of the black hole. However since the energyof the perturbation can leak into the black hole, these fluctuations are not normal modesand thus have been dubbed quasinormal. Their corresponding frequencies consist of areal and an imaginary part. As for the damped oscillator, the real part of the frequencyessentially determines the energy of the fluctuations, while the imaginary part is responsiblefor the damping. In AdS spacetimes the quasinormal modes satisfy the following boundaryconditions. At the horizon they are purely ingoing, whereas at the conformal AdS boundarythey have an asymptotic behavior that corresponds to a normalizable mode. In this paperwe determine the quasinormal spectrum of the D7-branes. The corresponding modes canbe grouped in terms of their transformation properties under spatial rotations of SO(3).We consider scalar modes given by perturbations of the brane embedding as well as vectormodes given by perturbations of the gauge field on the brane. As is well established bynow, the quasinormal frequencies of the dual gravity theory can be identified with the polesof correlation functions in dual thermal gauge theories [5, 6].

A system which is close to equilibrium can be described by linear response theory.There the effect of an external perturbation is given by the retarded two point function

– 6 –

folded against the source of the perturbation. By a Cauchy integration in the complexfrequency plane the response can be written as a sum over the contributions of the differentquasinormal modes (see e.g. [27]). Writing the time dependence as exp(−iωt) we note thata relaxation towards equilibrium can only happen if all the quasinormal modes lie in thelower complex half plane. Following [27,28] the response caused by an external perturbation(ω, k) can therefore be written as a sum over quasinormal frequencies

〈Φ(t, k)〉 = iθ(t)∑n

Rn (ωn, k) e−iΩnt−Γnt , (2.1)

where the quasinormal frequencies are ωn = Ωn − iΓn and their residues are Rn. If amode comes to lie in the upper half plane it results in an exponentially growing mode andtherefore represents an instability of the system. As we will see in the following, such aninstability does indeed occur in the scalar sector of the D7-brane fluctuations.

Of particular interest is the hydrodynamic limit that considers only fluctuations withsmall frequency and large wavelength. This hydrodynamic expansion can be seen as aneffective field theory where the degrees of freedom with large frequency and small wave-length are integrated out. In this effective field theory only the poles of correlation functionsclosest to the origin are important. More precisely, the modes with a dispersion relationobeying lim~k→0

ω(~k) = 0 represent the hydrodynamic regime. Hydrodynamic transportcoefficients such as the shear viscosity or the diffusion constants can be read off form thesepoles. Finally let us mention that recently it has been shown that the determinant of waveoperators in some asymptotically black hole backgrounds can be written in terms of thequasinormal modes [29].

3. Holographic Setup

We are interested in a large Nc thermal field theory, at finite baryon chemical potentialincluding fundamental and adjoint matter. Our matter content is that of the N = 4 superYang-Mills theory and a number Nf of N = 2 super Yang-Mills fundamental hypermulti-plets. We consider the quenched approximation with Nc →∞ and Nf Nc fixed. In thislimit the theory stays conformal at leading order in Nc.

The dual gravity setup is given by a stack of Nc D3-branes and Nf probe D7-branes.The metric generated by the D3-branes is a non-extremal AdS black hole backgroundplacing the field theory at finite temperature. The D7-branes allow for strings which haveone end on a D3- and the other end on the D7-brane. These 3-7 strings correspond toquarks in the fundamental representation. The length of these strings, i.e. the separationbetween the stacks of D3- and D7-branes corresponds to the quark mass Mq in the fieldtheory. On the world volume of the D7-branes we introduce a background gauge field A

which generates a chemical potential µ in the field theory at the boundary. Depending onthe problem it is convenient to work in different coordinate systems. We now introducethe ρ-coordinates and the z-coordinates.

– 7 –

3.1 The ρ-coordinate system

We find it convenient to use the ρ-coordinates of [16] to write the AdS black hole backgroundin Minkowski signature as

ds2 =%2

2R2

(−f

2

fdt2 + fd~x2

)+(R

%

)2

(d%2 + %2dΩ25) , (3.1)

with dΩ25 the metric of the unit 5-sphere and

f(%) = 1−%4H

%4, f(%) = 1 +

%4H

%4, (3.2)

where R is the AdS radius, with

R4 = 4πgsNc α′2 = 2λα′2 . (3.3)

This type of radial coordinate is better suited for calculation of the quasinormal modeswith the shooting method, where one places a cutoff at some large value of ρ.

The temperature of the black hole given by (3.1) may be determined by demandingregularity of the Euclidean section. It is given by

T =%HπR2

. (3.4)

In the following we may use the dimensionless coordinate ρ = %/%H , which covers the rangefrom the event horizon at ρ = 1 to the boundary of the AdS space at ρ→∞.

To write down the DBI action for the D7-branes, we introduce spherical coordinatesr,Ω3 in the 4567-directions and polar coordinates L, φ in the 89-directions [16]. Theangle between these two spaces is denoted by Θ (0 ≤ Θ ≤ π/2). The six-dimensional spacein the 456789-directions is given by

d%2 + %2dΩ25 = dr2 + r2dΩ2

3 + dL2 + L2dφ2

= d%2 + %2(dΘ2 + cos2 Θdφ2 + sin2 ΘdΩ23) ,

(3.5)

where r = % sin Θ, %2 = r2 + L2 and L = % cos Θ. Due to the symmetry, the embedding ofthe D7-branes only depends on the radial coordinate ρ. Defining χ = cos Θ, we parametrizethe embedding by χ = χ(ρ) and choose φ = 0 using the O(2) symmetry in the 89-direction.The induced metric G on the D7-brane probes is then

ds2(G) =%2

2R2

(−f

2

fdt2 + fd~x2

)+R2

%2

1− χ2 + %2(∂%χ)2

1− χ2d%2 +R2(1− χ2)dΩ2

3 . (3.6)

The square root of the determinant of G is given by

√−G =

√h3

4%3ff(1− χ2)

√1− χ2 + %2(∂%χ)2 , (3.7)

where h3 is the determinant of the 3-sphere metric.

– 8 –

3.2 Brane setup and background fields in z-coordinates

Here we introduce the coordinates also used in [13]. This set of coordinates maps thecompact interval [0, 1] to the distance between the conformal boundary and the black holehorizon. Such a compact radial coordinate is particularly well suited for the calculationof quasinormal modes using the relaxation method as explained in appendix B. We workin a non-extremal black-hole background generated by the stack of D3-branes giving theeffective metric

ds2

R2=

1z2

[−f(z)dt2 +

dz2

f(z)+ d~x2

]+ dΩ2

5, (3.8)

with f(z) = 1 − z4. Note that the horizon is located at z = 1, the AdS-boundary atz = 0. Frequencies and momenta measured in (3.8) are related to physical frequencies andmomenta by (ωph, kph) = πT (ω, k), where T is the temperature.

The D7-brane action reads

S =∫

d8ξ√−det(P [G] + 2πα′F ) , (3.9)

where P [G] is the metric’s pull-back on the Nf D7-branes. Here the induced metric isgiven by

ds2D7

R2= −f(z)

z2dt2 +

(1

z2f(z)−Θ′(z)

)dz2 +

1z2

d~x2 + sin2 Θ(z) dΩ23 , (3.10)

where Θ(z) describes the D7-brane embedding in the AdS-Schwarzschild background andthe embedding coordinates are xa = (t, ~x, z, α1, α2, α3). To second order in the fieldstrength the D7-brane Lagrangian can be written in the following way

L =√−P [G]

[1 + π2α′2FabF

ab]. (3.11)

With the zero order term, we obtain the background equation of motion

0 = 3 cos Θ(z)[−1 + z2(−1 + z4)Θ′(z)2] (3.12)

−z sin Θ(z)[(3 + z4)Θ′(z) + 2z2(1− z4)(2− z4)Θ′(z)3 − z(−1 + z4)Θ′′(z)] .

The brane embedding can be found by integrating this equation from the horizon out tothe boundary. As initial conditions one chooses χ0 = cos(Θ(1)) and demands regularityon the horizon. The quark mass can be read off from the behaviour at radial AdS infinitylocated at z = 0 [13]

Mq =12m√λT, m = χ′(z = 0) . (3.13)

We have plotted the mass parameter m as a function of χ0 = cos Θ(%H) in figure 2.The mass is not a single valued function of χ0. Although m is the physical parameter of thebrane embedding we prefer to quote χ0 values instead of m since much of our investigationswill take place in the regime where m ceases to be single valued. The change in the sign of∂m/∂χ0 is also suggestive of an instability. Indeed the maximum of m = 1.31 is reachedat χ0 = 0.962, which is precisely the value from where an unstable mode appears in thescalar sector of the quasinormal mode spectrum.

– 9 –

0.2 0.4 0.6 0.8 1.0Χ0

0.2

0.4

0.6

0.8

1.0

1.2

m

Figure 2: Plot of the dimensionless mass parameter m vs the cosine χ0 = cos Θ0 of the embeddingangle Θ(z) at the horizon, i.e. Θ0 = Θ(z = 1). The mass is not a single valued function of χ0 on[0, 1]! It takes a maximum value of m = 1.31 at χ0 = 0.962. The horizontal line indicates the firstorder phase transition at χ0 = 0.939.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5PSfrag repla ementsVSRR m0

~d = 0~d = 0:002~d = 0:1~d = 0:25Figure 3: At finite charge density: The dependence of the dimensionless quark mass m =2Mq/

√λT on the horizon value χ0 = limρ→1 χ of the embedding as shown in [30].

Looking at figure 3 it is obvious that at finite charge density the unstable regiondisappears at a critical density near dc = 0.00315 and for larger densities the quark massis a monotonously increasing function of the embedding parameter χ0, i.e. χ0 ∼Mq/T fordc ≥ 0.00315. This critical value will be confirmed in our analysis of the quasinormal modespectrum below.

The second order term produces the Maxwell equation

∂a(√−P [G]F ab) = 0, (3.14)

where Fab = ∂aAb − ∂bAa. We can choose the Az = 0 gauge and expand Aa in plane wavemodes. Moreover if we write the equations in a gauge invariant way using the electric fieldsin longitudinal EL = kphA0 + ωphA1 and transverse direction ~ET = ωph ~AT the equations

– 10 –

of motion are

E′′L(z) +[C(z) +

f ′(z)ω2

f(z)(ω2 − f(z)k2)

]E′L(z) +B(z)(ω2 − f(z)k2)EL(z) = 0 , (3.15)

E′′T (z) +[C(z) +

f ′(z)f(z)

]E′T (z) +B(z)(ω2 − f(z)k2)ET (z) = 0 , (3.16)

with

B(z) =1

f2(z)+z2Θ′(z)2

f(z),

C(z) = −1z

+ 2z(−2 + z4)Θ′(z)2 .

Our results are summarized in the following three sections.

4. Vanishing momentum and density

Our analysis produced a considerable amount of data and we will not show all of it becauseof some redundancy in the results. Technical details on our numerical methods are alsodeferred to the appendices A and B. The idea for this and the following two sectionsis, to have example figures for each case of interest and a listing of all the effects weobserve. We also provide a qualitative analysis by studying the correspondent Schrodingerequations. The discussion subsection in each of the cases is then devoted to the physicallymost interesting effects, i.e. the tachyon, diffusion mode, turning point.

4.1 Transverse vectors

The transverse equation of motion can be written in this simplified form

E′′T (z) +A1(z)E′T (z) +B(z)(ω2 − f(z)k2)ET (z) = 0 , (4.1)

where A1(z) = C(z) + f ′(z)f(z) . Close to the boundary of AdS (z → 0+), the differential

equation reduces to

E′′T (z)− 1zE′T (z) = 0 , (4.2)

which has the solution ET (z) = A+Bz2. According the dictionary of the correspondence,A should be zero in order to study the quasinormal states. Close to the horizon (z → 1−),the differential equation is given by

E′′T (z) +1

z − 1E′T (z) +

ω2

16(z − 1)2ET (z) = 0 , (4.3)

with the solution ET (z) = A′(1− z)iω/4 +B′(1− z)−iω/4. The infalling boundary conditionis fulfilled by choosing A′ = 0.

At this point we can perform the next transformation ET (z) = (1−z)−iω/4y(z), in orderto split the infalling singular part from the regular part of the function. In consequence

– 11 –

the function y(z) must satisfy the boundary conditions y(0) = 0 and y(1) = 1 and thedifferential equation turns out to be

y′′t (z) + [α1 + iωγ1]y′t(z) + [α0 + iωβ1 + ω2β2]yt(z) = 0 , (4.4)

with α1 = A1, γ1 = 12(1−z) , α0 = −k2f(z)B(z), β1 = 1+A1(1−z)

4(1−z)2 and β2 = − 116(1−z)2 +B(z).

Results for the quasinormal modes of the transverse vectors in this case are shown infigure 4.

4.2 Longitudinal vectors

The equation of motion for longitudinal vectors is given by

E′′L(z) +[A1(z)(ω2 − f(z)k2) + C0(z)

ω2 − f(z)k2

]E′L(z) +B(z)(ω2 − f(z)k2)EL(z) = 0, (4.5)

with C0(z) = k2f(z). The asymptotic behavior of this equation is the same for the trans-verse e.o.m, then if we do the same transformation that above, we obtain this equation:

y′′l (z) +[α′1 + α′2ω

2

ω2 − k2f(z)+ iωγ′1

]y′l(z) +

α′0 + β′2ω2 + β′3ω

4 + i(ωβ′1 + ω3β′4)ω2 − k2f(z)

yl(z) = 0, (4.6)

with

α′1 = C0(z)−A1(z)k2f(z) , α′2 = A1(z) ,

γ′1 =1

2(1− z), α′0 = k4f2(z)B(z) ,

β′1 =C0(1− z) + k2(A1(z − 1)− 1)f(z)

4(z − 1)2, β′3 = B(z)− 1

16(1− z)2,

β′2 =k2(1− 32(−1 + z)2B(z)

)f(z)

16(−1 + z)2, β′4 =

1 +A1(z)(1− z)4(−1 + z)2

.

In the case with k = 0 the differential equations for transverse and longitudinal fluc-tuations are the same, in consequence their quasinormal spectra coincide, see figure 4.

Results for vectors Figure 4 shows the first and second QNM in the complex frequencyplane. Starting with zero quark mass, i.e. at high temperature (red color), the imaginarypart monotonously decreases with decreasing temperature. This means the correspondingmode becomes more and more stable. In contrast to that the real part of the quasinormalfrequency first grows until it reaches a maximum and then decreases as well with decreasingtemperature. This maximum in the real part of the QNM lies above the meson meltingtransition (indicated by a short horizontal dash). The melting transition takes place at acritical angle χ0 = 0.939 from which on the Minkowski embeddings are thermodynamicallyfavored, not the black hole embeddings. We have chosen to remain in the so-called under-cooled phase keeping the black hole embeddings even beyond the transition. This phaseis accessible since the meson-melting is a first order transition. So the undercooled phaseis metastable. However we will see in the following section that at a smaller temperature

– 12 –

----ÈÈ1.7 1.8 1.9 2.0 2.1 2.2

ReHΩL

0.5

1.0

1.5

2.0

-ImHΩL

----ÈÈ2.0 2.5 3.0 3.5 4.0

ReHΩL

1

2

3

4

-ImHΩL

Figure 4: Location of the first (left) and second (right) quasinormal modes in the complexfrequency plane for the vector fluctuations at vanishing momentum (k = 0) as a function of theembedding χ0. Red color indicates small quark mass, or high temperature, while the temperaturedecreases towards blue colors. The horizontal (black) dash indicates the frequency at the first orderphase transition where the angle is χ0 = 0.939. The vertical (red) dash indicates the frequecy atwhich the embeddings become locally unstable at χ0 = 0.962. The modes are followed down toembeddings with χ0 = 0.999875.

relaxation shootingχ0 Reω Imω Reω Imω

1st QNM0 2.0000 -2.0000 2.0000 -2.0000

0.48 2.1075 -1.5973 2.1075 -1.59720.92 2.0656 -0.4853 2.0657 -0.4852

2nd QNM0 4.0054 -3.9976 3.9995 -4.0004

0.48 4.0417 -3.3366 3.9324 -3.33860.92 3.4397 -1.3093 3.4397 -1.3093

Table 1: Exemplary values for the first and second vector QNM frequencies at k = 0 for differentvalues of χ0 parametrizing the D7-embedding. See figure 2 for the relation between χ0 and thequark mass Mq. The first pair of values in each row is obtained from the relaxation method, thesecond pair stems from requiring the shooting solution to vanish at the AdS boundary. We find aremarkable agreement.

below the melting transition, i.e. a larger angle χtachyon0 ≈ 0.962, this undercooled phase is

destabilized by the scalar fluctuation becoming tachyonic. In the figures this is indecatedby a red vertical dash.

Exemplary numerical values for the vector QNM frequencies at k = 0 are given intable 1. We find a remarkable agreement between the values obtained with two differentmethods: the relaxation method and the shooting method. The results are in good agree-ment for all parameter regions and in all the cases we treat in this work. Therefore weexclusively show results produced with the relaxation method from now on.

– 13 –

1.90 1.95 2.00 2.05 2.10 2.15 2.20

0.5

1.0

1.5

2.0

ReHΩL

-ImHΩL

Shooting

Relaxation

Figure 5: Shoot and relax: Comparison of the shooting method result (green squares) with therelaxation method results (black circles) for the location of the first transverse vector quasinormalmode at vanishing momentum k = 0, density d = 0. Along the curves the temperature is varied.Our two methods agree very well.

4.3 Scalar

In the DBI-action (3.9) we let the Θ-angle fluctuate and we split this fluctuation δΘ into aproduct of its singular and regular parts δΘ(z) = (1− z)−iω/4zy(z). With this change theinfalling boundary condition at the horizon is translated into y(1) = 15 and the Dirichletcondition at the boundary implies y(0) = 0. The equation of motion for scalar fluctuationsthen reads

y′′(z) + [a1(z) + ic1(z)ω] y′(z) +[a0(z) + ib1(z)ω + b2(z)ω2

]y(z), (4.7)

with

a1(z) = A1(z) +2z, a0(z) =

A1(z)z

+A0(z)−B(z)2k2f(z) ,

c1(z) =1

2(1− z), b2(z) = B(z)2 − 1

16(1− z)2,

b1(z) = −(1−A1(z)(1− z))z − 24z(1− z)2

.

Results for the scalar The first and second scalar QNM at vanishing momentum can befound in figure 6. The basic behavior is similar to that of the vector modes. Increasing thequark mass from zero the real part of the QNM frequency again shows a turning behaviormoving first to larger values, then to smaller values of Re(ω). However, in contrast to thevectors, the scalar QNM frequency also shows a turning behavior in the imaginary partIm(ω). This means that increasing the quark mass, i.e. decreasing the temperature, thecorresponding modes first decay faster, then beyond the turning point they decay slower and

5Notice that the equation for y(z) is linear and that we can scale y by and arbitrary constant, therefore

we can always choose the boundary condition at the horizon to be y = 1.

– 14 –

----ÈÈ

0.5 1.0 1.5 2.0 2.5ReHΩL

0.5

1.0

1.5

2.0

-ImHΩL

----ÈÈ

2.0 2.5 3.0 3.5 4.0 4.5ReHΩL

1

2

3

4

-ImHΩL

Figure 6: Location of the first and second quasinormal modes in the complex frequency planefor the scalar fluctuations at vanishing momentum (k = 0) as a function of the embedding χ0. Redcolor indicates small quark mass, or high temperature, while the temperature decreases towardsblue colors. The horizontal (black) dash indicates the frequency at the phase transition wherethe angle is χ0 = 0.939 whereas teh vertical (red) dash indicates the onset of the instability atχ0 = 0.962.

slower as the mode approaches the real axis. Moreover the scalar QNMs do not asymptoteto the real axis as fast as the vector QNMs do. Instead the scalar QNM frequencies evenat large masses still have a considerable imaginary part of roughly 1/2. All the values forthe scalar modes are excellent agreement with the ones obtained previously in [13].

The short dash in the figures again shows the location of the known meson meltingtransition where the initial angle is χ0 = 0.939. Increasing the mass further while staying inthe black hole phase, we observe a scalar QNM to become tachyonic. This point is markedby a vertcal dash in the figures. Figure 7 shows the appearance of an unstable modeexplicitely. This particular mode is special since it has vanishing real part but it startswith an extra-ordinary large imaginary part of the QNM frequency at zero quark mass.Increasing the quark mass this purely damped mode moves closer to the real axis until itcrosses to become unstable at χ0 = 0.962 corresponding to the maximal mass for blackhole embeddings of m ≈ 1.31. This particular mode has not been observed in previousstudies because at vanishing quark mass it is located very deep in the complex frequencyplane near Imω ≈ −8, while for example the first scalar QNM has Imω = −2 at vanishingquark mass. In principle there could be an infinite tower of such purely imaginary modes,each crossing the real axis at the larger and larger quark mass. However, the accuracyof our numerics proved insufficient to establish additional modes beyond this lowest one.In any case once this mode has crossed the real axis the D7 brane embedding is locallyunstable and can not be taken as a (metastable) ground state. This raises the question ofwhat is the true ground state in this regime. It might be that there is another type of D7-brane embedding that is reached somehow by condensation of the scalar mode. Anotherpossibility is that there is simply no locally stable embedding beyond that point.

4.4 Schrodinger potential analysis

In this section we present a qualitative analysis of the quasinormal spectrum using the

– 15 –

0.2 0.4 0.6 0.8 1.0Χ0

-6

-4

-2

0

ImHΩL

0.92 0.94 0.96 0.98 1.00Χ0

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3ImHΩL

Figure 7: Left: The plot shows a purely imaginary quasinormal mode at k = 0 as a functionof the embedding. Right: Zooming into the region where the scalar mode crosses the real axisbecoming tachyonic approximately at χ0 = 0.96221

fact that the equations of motion for the fluctuations can be rewritten in the form of theSchrodinger equation (see Appendix C for more details)

−∂2R∗ψ + VSψ = Eψ , (4.8)

where R∗ is a tortoise-like coordinate. The Schrodinger potential VS determines the energyspectrum E which is related to the quasinormal spectrum by E = ω2.

At zero baryon density and zero momentum, the potentials for the vector and scalarmodes are already discussed in [25] and [14], respectively. Nevertheless we include thediscussion here for completeness. In fig. 8 we present the Schrodinger potential for thevector and scalar fluctuations at different quark masses parametrized by χ0. In these plotswe observe an infinite wall in the potential at R∗ = 0 which corresponds to the AdSboundary. In addition to this wall the potential for the vector modes develops a step-shape as we increase the quark mass. In [25] it is shown that the imaginary part of thequasinormal frequency decreases as the step gets longer which is consistent with our resultfound in fig. 4.

For the scalar modes a negative well arises in the Schrodinger potential. This wellbecomes deeper and wider as we increase the quark mass and therefore support a ‘bound’state with E < 0 which corresponds to a tachyonic quasinormal frequency Im ω > 0. Thiswell and the ‘bound’ state are studied in [14]. The Schrodinger analysis clearly shows theexistence of a tachyonic mode which we already found in fig. 7.

4.5 Discussion: Tachyon and De-singularization

In this section we discuss why the so-called undercooled phase shows unphysical mesonspectra which do not approach the known ones in the supersymmetric limit. Further wediscuss that finite density cures this behavior by de-singularizing the geometry, i. e. bysmoothing out the limiting embedding.

As mentioned above the scalar fluctuation becomes tachyonic once the quark massparameter has reached its maximum as a function of χ0. It is not to be expected thatthe region beyond that point contains physically relevant or meaningful signatures. Thisregion contains the limiting embedding which only touches the horizon and geometrically

– 16 –

0 1 2 3 4 5

R*0

1

2

3

4

5

PSfrag repla ementsVS

R(a)

1 2 3 4

R*

-2

-1

0

1

2

3

4

5

PSfrag repla ementsVS

R(b)

Figure 8: Schrodinger potential of the vector (a) and scalar (b) fluctuations for different values ofχ0. The different colors correspond to χ0 = 0 (black), 0.5 (green), 0.9 (blue), 0.95 (orange), 0.99(red).

separates Minkowski from black hole embeddings. Here meson spectra had been studiedearlier [14, 25]. These meson spectra display a singular behavior in the sense that all thequasinormal modes (first, second, . . . ) approach one single ’attractor’ energy (or frequency)near the limiting embedding. The geometric reason for this is the scaling symmetry forthe embeddings in the near-critical region [12]. That scaling symmetry implies that nearthe critical quark mass (or temperature) there exists no preferred scale on the brane. Inthis sense there is no scale which could determine the distance between resonances in thebrane fluctuations, i.e. between the distinct meson mass resonances, or quasinormal modesequivalently.

At finite fixed baryon density however, this particular scaling symmetry is broken [16].Therefore the chemical potential introduced together with that density does set the scalefor the separation between the distinct quasinormal modes. This is somewhat analogousto the behavior of the thermal quasinormal mode spectrum when the temperature of thedual black hole background is lowered towards zero. In that case the quasinormal modesare known to approach one single value, i.e. the spectrum becomes singular in our sense.There the scale which determines the distance between distinct excitations is clearly thetemperature since it is the only scale in the theory. In the zero temperature limit this scalevanishes and the scaling symmetry is restored.

As observed earlier [30] the meson spectrum at finite density approaches the supersym-metric one at low temperatures. In this sense the theory is de-singularized by finite baryondensity. We will see this explicitly in the quasinormal mode spectra at finite density insection 6.

5. Finite momentum but vanishing density

We now turn to the case of non-zero spatial momentum.

5.1 Transverse vectors

We obtain results at finite momentum from the numerical solution of equation (4.1) withdistinct non-zero values of k.

– 17 –

----

--

--

2.0 2.5 3.0 3.5 4.0 4.5 5.0ReHΩL

0.5

1.0

1.5

2.0

-ImHΩL

k=4

k=3

k=2

k=1-- -- -- --

2 3 4 5 6ReHΩL

0.5

1.0

1.5

-ImHΩL

k=5.0

k=4.8

k=4.4

k=4.0

2.0 2.5 3.0 3.5 4.0 4.5 5.0ReHΩL

0.5

1.0

1.5

-ImHΩL

k=4.20

k=4.16

k=4.08

k=4.00

--

5.3 5.4 5.5 5.6 5.7 5.8ReHΩL

1.25

1.30

1.35

1.40

1.45

1.50

-ImHΩL

k=5

Figure 9: First quasinormal mode for transverse vector fluctuations at different spatial momentak. Horizontal short dashes across the curves show the location of the meson melting transition. Atk ≈ 4.16 a spiral structure appears and the curves also asymptote to a distinct small temperaturevalue from that k on. Along the curves the quark mass parameter χ0 = cos Θ0 is varied. See figure 2for its relation to the quark mass and temperature.

Figure 9 shows the first of the transverse vector QNMs at different values of the spatialmomentum k. The behavior in the region k = 0, . . . , 4 is very similar to the k = 0 case.Although its trajectory in the complex frequency plane becomes more wavy at larger k, thefirst scalar QNM still starts at quite large real and imaginary parts in order to approachthe real axis and smaller real parts when temperature is decreased. Distinct curves fordifferent values of k within numerical accuracy approach a single limiting value ω0 at smalltemperatures. It is interesting to note that the turning point in Re(ω) mentioned in theprevious section for k = 0 disappears when k reaches values between k = 1 and k = 2.

A quite substantial qualitative change appears at k = 4.16, where the trajectory ofthe mode in the frequency plane develops one loop. Also the trajectories at higher k havethis looping behavior. At the same time these curves with one loop do asymptote to asingle small temperature frequency value ω1 as well. But this limiting value is distinctfrom the limiting value which is approached by the low k curves without the loop, i.e.ω1 6= ω0. This fact suggests that the loop-behavior and the distinct limiting value aresomehow related. For the longitudinal vector fluctuations (see figure 11) we will explicitlysee that this relation generalizes to all fluctuations and to higher loops in this way: Allthe first QNM trajectories for scalar and vector fluctuations with k < kn have n loops and

– 18 –

they asymptote to a small temperature limit frequency ωn (within numerical accuracy)with ωn+1 > ωn. Note that the loop behavior appears in a physical, thermodynamicallypreferred phase, i.e. before the meson melting transition and also way before the tachyonappears.

The loops are absent in the second quasinormal mode as seen from figure 35. Never-theless the second quasinormal mode also asymptotes to distinct low temperature limitsωn above distinct certain momenta kn.

1 2 3 4 5k

2.5

3.0

3.5

4.0

4.5

5.0

5.5

ReHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

1 2 3 4 5k

-1.8

-1.6

-1.4

-1.2

ImHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

Figure 10: Dispersion relation for the first transverse quasinormal mode at distinct quark masses,or equivalently temperatures, parametrized by the embedding parameter at the horizon, χ0. Seefigure 2 for the relation between χ0, the temperature and quark mass Mq.

Figure 10 captures the dispersion relation of the first transverse vector QNM at differ-ent values of χ0.

5.2 Longitudinal vectors

We obtain results at finite momentum from the numerical solution of equation (4.5) withdistinct non-zero values of k.

The behavior of the longitudinal vector QNMs is qualitatively similar to that of thetransverse ones discussed in section 5.1. The mentioned loops in the frequency planetrajectory do appear at smaller values k ≈ 3.4 in the longitudinal channel than they do inthe transverse one. However, while in the transverse vector case the loops appeared beforethe meson melting transition, in the longitudinal case the transition takes place before thefirst loop is terminated as can be seen from the figure 11. Just as for the transverse vectors,also the second QNM of the longitudinal vectors does not have any loops in its complexfrequency plane trajectory, as figure 37 shows.

The dispersion relations for the first and second QNM of the longitudinal vector fluc-tuation are depicted in figure 12 and figure 38 in appendix E, respectively.

5.3 Scalar

We obtain results at finite momentum from the numerical solution of equation (4.7) withdistinct non-zero values of k.

The general behavior of the scalar QNMs is qualitatively similar to that of the longi-tudinal vector QNMs. Figure 13 shows the first of the scalar QNMs at momenta between

– 19 –

-- ---- --

--

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5ReHΩL

0.5

1.0

1.5

2.0

-ImHΩL

k=5

k=4

k=3

k=2

k=1

-- -- --

2.0 2.5 3.0 3.5 4.0ReHΩL

0.2

0.4

0.6

0.8

-ImHΩL

k=3.4

k=3.2

k=3.0

-- -- --

2.5 3.0 3.5 4.0 4.5 5.0ReHΩL

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-ImHΩL

k=4.0

k=3.8

k=3.4

--

5.42 5.44 5.46 5.48 5.50 5.52ReHΩL

0.52

0.54

0.56

0.58

0.60

0.62

0.64

-ImHΩL

k=5

Figure 11: The first quasinormal mode for longitudinal vector fluctuations at distinct values ofk. Along the curves the quark mass parameter χ0 = cos Θ0 is varied. See figure 2 for its relationto the quark mass and temperature.

1 2 3 4 5k

2.5

3.0

3.5

4.0

4.5

5.0

5.5ReHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

1 2 3 4 5k

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6ImHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

Figure 12: Dispersion relation for the first quasinormal longitudinal vector mode fluctuation atdistinct values of the quark mass parameter χ0 (see figure 2 for its relation to the quark mass atvanishing density).

k = 1 and k = 5. Also in this case the overall behavior is that the real and imaginaryparts decrease as temperature is decreased along the curves. At small k, e.g. k = 1, thereis a turning point present in the real as well as in the imaginary part. These turningpoints again disappear between k = 1 and k = 3. Just like for the first of the longitudinalvector QNMs multiple loops form successively for larger values of k. The meson meltingtransition appears before the first of the loops has terminated. Again the number of loopsseems to be directly related to the low temperature value ωn to which the curves for allk > kn asymptote. Dispersion relations for the first scalar QNM are shown in figure 14.

– 20 –

The corresponding figures for the second scalar QNM are figure 39 and 40 in appendix E.Just like for the vectors there are also no loops in the second QNM for the scalars.

-- --

--

--

--

1 2 3 4 5 6ReHΩL

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

-ImHΩL

k=5k=4k=3k=2k=1 --

---- --

4.5 5.0 5.5ReHΩL

1.1

1.3

1.4

1.5

1.6

1.7

-ImHΩL

k=5k=4.8k=4.4k=4

Figure 13: First scalar quasinormal mode at distinct momenta k. Along the curves the quark massparameter χ0 = cos Θ0 is varied. See figure 2 for its relation to the quark mass and temperature.

1 2 3 4 5k

3.0

3.5

4.0

4.5

5.0

5.5

ImHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

1 2 3 4 5k

-1.8

-1.7

-1.6

-1.5

-1.4

-1.3

ImHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

Figure 14: Dispersion relation for the first scalar quasinormal mode at distinct values for thequark mass parameter χ0 (see figure 2 for its relation to the quark mass at vanishing density). Thereal part of the QNM frequency is shown on the left, the imaginary part on the right.

5.4 Schrodinger potential analysis

Just as in the previous section at zero density and momentum, we here compute theeffective potential for the scalar and vector fluctuation equations (4.7) and (4.5), (4.1) (seeAppendix C for more details).

We begin by examining the Schrodinger potential for the scalar fluctuations in figure 15.The lowest (red) curve shows the potential at vanishing momentum and density at χ0 =0.9999. That is near the limiting embedding, far beyond the thermodynamic transition toMinkowski embeddings and far beyond the appearance of the tachyon in the spectrum. Theresults at smaller χ0 are qualitatively the same, but we plot large χ0 in order to investigatethe tachyon and the reason for the different ”attractor” frequencies in that large massregime. The scalar potential clearly exhibits a wide negative dip in which the tachyonicscalar mode resides, compare figure 7. As the momentum is increased the potential is

– 21 –

3 4 5 6

-1

1

2

3

4

PSfrag repla ementsVS

R RFigure 15: The scalar Schrodinger potential Vs (zooming in on the minimum of the po-tential) versus the radial coordinate R∗ defined in section 4.4 at increasing momenta k =0, 2, 4, 6, 8, 10, 12, 14, 16 from bottom to top curve with the quark mass parameter χ0 = 0.9999(see figure 2 for its relation to the quark mass at vanishing density), d = 0. The dip supporting thetachyon narrows.

lifted and the negative dip is narrowed. In this way the lowest possible excitation is pushedtowards more positive energy values becoming non-tachyonic at large momenta. Howeverthe theory is clearly unstable against condensation of the scalar fluctuations already atχ0 ≥ 0.962 and k = 0. In figure 16 we zoom out to larger values of the potential. Forincreasing momentum, a step forms near the boundary. When comparing to figure 8 (a)we see that the scalar potential at finite momentum is similar to the vector case at zeromomentum. In figure 16 the step becomes higher and longer for increasing momentum,while its plateau becomes shorter, i.e. most of the plateau is located near the boundary.Therefore conceptually the analysis of [25] as discussed in section 4.4 applies as in the vectorcase: When the step becomes longer, the imaginary part of the quasinormal frequenciesdecreases. This is consistent with our observations in figure 13 (compare for example theinitial points where χ0 = 0 for increasing momenta from curve to curve). The increasingreal part of the quasinormal frequency observed in figure 13 is due to the fact that thepotential step rises closer to the horizon at larger momentum. This narrows the potentialand rises the excitation energies.

Turning now to the transverse vector fluctuations, we observe a step potential in fig-ure 17. The larger the momentum, the earlier the potential rises towards infinity whenapproaching the boundary at R∗ = 0. So effectively the boundary moves towards thehorizon and the length of the plateau of the potential step becomes shorter. This is differ-ent from the scalar modes discussed above. While the imaginary part of the quasinormal

– 22 –

1 2 3 4 5 6 7 8

20

40

60

80

100

120

PSfrag repla ementsVS

R RFigure 16: The scalar Schrodinger potential Vs (zooming out to larger values of the potential)versus the radial coordinate R∗ at increasing momenta k = 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 40from bottom to top curve with quark mass parameter χ0 = 0.9999 (see figure 2 for its relation tothe quark mass at vanishing density), d = 0. A step forms towards the boundary, similar as thatfor the vectors. The grid line at R∗ = 1 serves to guide the eye only.

frequency decreases for increasing momentum just as in the scalar case, the Schrodingerpotential shows a different behavior: For increasing momentum, the Schrodinger potentialapproaches the shape of a wall. Where as in the scalar case, the formation of the step isresponsible for lowering the imaginary part of the quasinormal frequency, here we expectthat the formation of the wall is responsible for a similar decrease of the imaginary part ofthe quasinormal frequency.

5.5 Discussion: Breakdown of hydrodynamics, ’attractors’ and the tachyon

In this discussion we focus on three distinct physical implications of the quasinormal modesdescribed previously in this section. The longitudinal vector modes tell us when the hydro-dynamic approximation breaks down, while the novel purely imaginary scalar mode rendersthe whole theory unstable as it becomes tachyonic. Both scalar and vector modes asymp-tote to ’attractor’ frequencies. This behavior is probably related to a spiraling behavior ofthe quasinormal mode’s trajectories with changing quark mass or temperature.

5.5.1 Hydrodynamics to Collisionless Crossover

Contrary to the transverse vector and scalar channel the longitudinal vector channel alsohas a hydrodynamic quasinormal mode, i.e. a mode whose dispersion relation does notshow a gap at zero momentum, limk→ ω(k) = 0 (see figure 18, left). This mode representsdiffusion of baryon charge. It is a mode whose frequency is purely imaginary and therefore

– 23 –

1 2 3 4 5 6

1

2

3

4

5

PSfrag repla ementsVS

R RFigure 17: The transverse vector Schrodinger potential Vs versus the radial coordinate R∗ atincreasing momenta k = 0, 2, 4, 10, 20, 40 from bottom to top curve with quark mass parameterχ0 = 0.9999 (see figure 2 for its relation to the quark mass at vanishing density), d = 0. Thepotential step becomes shorter since the potential rises farther and farther away from the boundaryR∗ = 0.

results in a purely damped time evolution without any oscillation. At small momentumthe dispersion relation is well approximated by the diffusion kernel ω = −iDk2, as seenin figure 18 on the left. Fitting our numerical data to this we can extract the diffusionconstant D. Since it has been calculated before in the literature in [14] we do not restatethis result. We checked however that our values are consistent with the results there.

On general grounds one expects that a many body system shows a crossover from hy-drodynamic behavior at long wavelengths to a coherent or collisionless behaviour at smallwavelengths. In the holographic context this has first been discussed in an AdS4 examplein [31] by calculating spectral functions. A more direct way to see this crossover can beobtained by studying the quasinormal mode spectrum. At small momentum the hydrody-namic mode should be the dominant one, i.e. the one with the largest imaginary part. Atsmall wavelength we expect the dominant modes to have frequencies whose imaginary partis much smaller than their real parts. This means that at large wavelength the responsewould be simply an exponential decay whereas at short wavelength the response would bea slowly decaying oscillation. In terms of the quasinormal mode spectrum this implies thatthe purely imaginary diffusion mode as a function of momentum has to cross the imagi-nary part of the dispersion relation for the lower non hydrodynamic modes. Indeed thisis what happens for the R-charge and momentum diffusion in the strongly coupled N = 4theory as discussed in [27, 28]. We therefore define the crossover from the hydrodynamicregime to the collisionless regime through the momentum at which the imaginary part of

– 24 –

the lowest non-hydrodynamic mode crosses the purely imaginary diffusion mode. Fromthat wavelength on it is the lowest gaped quasinormal mode which dominates the late timeresponse6. It should be mentioned that there is at least one other way of how this crossovercan be established in terms of quasinormal modes. It could also be that the purely imagi-nary diffusion mode pairs up with another purely imaginary but non-hydrodynamic modewhich allows them to develop real parts as well and to move off the imaginary axis. Thisseems to be the preferred mechanism for AdS4 black holes [33] and it also appears on probeD-branes representing defects in the four-dimensional strongly coupled CFT [34]. As willbe shown in a companion publication this is also what happens if finite baryon density isintroduced [35]. This crossover has recently also been investigated for AdS black holes invarious dimension in the time domain in [36].

We have numerically determined the crossover point defined above for different em-beddings. As seen from the right side of figure 18, the crossover moves to higher momentaas the embedding angle χ0 is increased. This means that the brane responds to baryoncharge fluctuations in a purely absorptive way for smaller and smaller wavelengths as thequark mass is increased, or equivalently as the temperature is decreased. Of course theactual rate of absorption given by the absolute value of the imaginary part of the frequen-cies decreases with decreasing temperature. Nevertheless it is somewhat surprising thatthe crossover towards the collisionless regime takes place at smaller wavelengths for lowertemperatures.

5.5.2 ”Attractor” frequencies

Here we briefly discuss the appearance of spiraling QNM trajectories and their relation tothe ”attractor” frequencies found in section 5.

First we should note that in most cases the spirals in the QNM trajectories occur beforethe scalar in the spectrum becomes tachyonic. Thus the spiral is a physical signature on thestable or metastable branch of the theory. However the ”attractor” frequencies to whichthe trajectories asymptote at large quark mass parameters χ0 → 1 are located deep in theunstable phase of the theory. Therefore the ”attractor” frequency is no signature of thephysical stable sector of the theory. Nevertheless it would be interesting to understand theapparent direct relation of these ”attractor” frequencies and the number of spirals in theQNM trajectories since the spirals are physical, as noted before.

6There is a small puzzle related to that. If the prolongation of the hydrodynamic mode to large momenta

is constantly increasing, the front velocity computed from it seems to violate causality. As is well-known

already the diffusion kernel violates causality because of the k2 behaviour and the extension to larger

momenta shows even higher exponents in the dependence on k. So how manages the theory to preserve

causality? The resolution has been presented in [27]: the residues of the diffusive quasinormal mode vanish

for large momenta and therefore this mode ceases to exist in the dangerous limit of large k. On the other

hand one can study the hydrodynamics by fixing a real frequency and then looking for complex roots

in the momentum k as it has been done in [32]. The quasinormal frequencies or the complex momenta

respectively are roots of infinite order polynomials (or at least of extremely high order polynomials in a

truncated approximation of the holographic Green function). Therefore it is not possible to simple infer

the behaviour of the complex momentum modes from the quasinormal modes. Indeed as shown in [32] the

front velocity of hydrodynamic mode as calculated from the complex momentum roots behaves perfectly

causal and approaches 1 to very good numerical accuracy.

– 25 –

0.2 0.4 0.6 0.8 1.0 1.2 1.4k

-1.5

-1.0

-0.5

0.0

ImHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

1.3 1.4 1.5 1.6 1.7k

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6ImHΩL

Χ0 = 0.72

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0

Figure 18: Left: The dispersion relation for the diffusion mode at distinct values for themass/temperature parameter χ0 (see figure 2 for its relation to the quark mass at vanishing den-sity). Right: Intersection of the diffusive mode with the imaginary part of the first longitudinalquasinormal mode. The intersection point moves to larger values of k, but to smaller values ofIm(ω) as the mass/temperature parameter χ0 is increased.

Note that a spiraling behavior for changing the temperature has been observed in thequark condensate in this system for near-limiting brane embeddings in [12]. There thespirals are due to oscillations of the parameters of the embedding, i. e. the quark massand the quark condensate. In particular the asymptotic value m oscillates. This behaviorhas only been observed in the near-limiting embeddings. In contrast to that our loops inthe QNM-trajectories appear way above the critical embedding already. Nevertheless, asstated before there is an apparent connection between the number of loops in our QNM-trajectories and the near-limiting ’attractor’ frequency. In this way we could argue thatat finite momentum we see the near-limiting embedding oscillations reflected already inthe non-critical region in spiraling QNM-trajectories. In other words both the QNM-loopsand the spiraling quark condensate might have the same origin, namely the oscillatingembedding parameters which are related to the aforementioned scaling symmetry of thenear-limiting embedding [12].

Unfortunately the Schrodinger potentials at finite momentum but vanishing densityclose to χ0 = 1 in figure 15 and 17 do not show any distinct feature hinting neither ondiscrete special frequencies ωn nor on the jumps between them at critical momenta.

5.5.3 Tachyon: A new hydrodynamic mode

We briefly discuss here the behavior of the scalar mode becoming tachyonic as explainedin the previous section. This mode turns into a hydrodynamic mode in a special case.

As expected this scalar mode becomes tachyonic at higher and higher values for thequark mass parameter χ0 as the momentum of the excitation is increased. In the parame-ter space spanned by charge density, temperature and quark mass there is one interestingspecial point: that is the location χcrit0 (d) where the scalar mode becomes tachyonic. Justat this special quark mass/temperature value, this scalar mode develops a hydrodynamicdispersion relation, i.e. limk→0 ω → 0. In other words the scalar mode which shows theinstability of the system turns into a hydrodynamic mode just at the critical point. Thiscould signal that there is a transition to a new phase. For instance this transition might

– 26 –

be similar to the glass transition in supercooled liquids discussed e. g. in [37]. Lastly theremight not exist a new stable ground state since we might be scanning a regime where nostable brane embedding exists besides the thermodynamically preferred Minkowski embed-ding.

6. Finite density but vanishing momentum

In this section we turn back to zero momentum, but switch on a finite baryon density andchemical potential. The non-normalizable mode of the zero component of the gauge fieldliving on the D7-brane gives the chemical potential in the field theory. Working in the ρcoordinates we define

limρ→∞

A0(ρ) = µ . (6.1)

In order to study vector mesons in this background we consider fluctuations of the gaugefield about this background field. The equation of motion for the zero component of thegauge field thus reads [16]

∂ρA0 = 2df√

1− χ2 + ρ2χ′2√f(1− χ2)

[ρ6f3(1− χ2)3 + 8d2

] . (6.2)

Since the asymptotic behavior for the embedding function χ is known we can also give theasymptotics for the gauge field at infinity

A0 = µ− 1ρ2

d

2πα′+ . . . , (6.3)

where µ is, as already mentioned, the chemical potential and d parametrizes the baryonnumber density nB by

d =2

52nB

Nf

√λT 3

. (6.4)

We divided this chapter into four sections in the same manner as the former ones,except that we do not have to handle longitudinal vectors here since k = 0. But we willinvestigate the transverse vectors and the scalar modes.

Then we will describe our results, especially the tachyon behavior and the turningpoints of the meson mass. Furthermore we present a Schrodinger potential analysis tounderstand the above results in a different way. We investigate the turning behavior of thequasinormal frequencies further by an analytical calculation close to the horizon at largefrequencies.

6.1 Transverse vectors

To obtain the equation of motion for the transverse vector modes, we have to vary the DBIaction with respect to the gauge field fluctuations about the background and retain these

– 27 –

terms up to second order [30]. The equation of motion for the fluctuations of the gaugefield reads

0 = E′′ + ∂ρ log

(f

f

√1− χ2

√ρ6f3(1− χ2)3 + 8d2

f(1− χ2 + ρ2χ′2)

)E′ + 2ω2 f

f2

1− χ2 + ρ2χ′2

ρ4(1− χ2)E . (6.5)

The connection between the gauge invariant field E and the gauge field is simply E = ωphAi,where we are free to choose any Minkowski spatial direction i = 1, 2, 3. Furthermorethe parameter ω is a dimensionless frequency defined as ω = ωph/πT with the physicalfrequency ωph. A more detailed derivation of this equation of motion can be found in [30].

The solution to this equation of motion can be obtained by numerical methods, de-scribed in appendix A. Then we can compute the correlator in the complex plane wherethe quasinormal modes appear as poles. We aim for an explanation of the turning behaviorof the meson mass found in [30]:At large quark mass the meson mass increases proportional to the quark mass Mq as ex-pected from the formula found in the supersymmetric case for vector and scalar mesonmasses (for vanishing angular momentum l = 0 on the S3) [8]

Mmeson =2Mq√λTR2

√2(n+ 1)(n+ 2) , n ∈ N . (6.6)

Whereas for small quark mass there exists a region where the meson mass is decreasingwhen the quark mass is raised.

Results for transverse vectors Figure 19 (a) displays the paths of the first quasinormalmode in the complex frequency plane for different densities. The paths are parametrized bythe quark mass over temperature ratio. When the quark mass is zero, i. e. our embeddingis flat, the quasinormal mode is located near the point 2(1− i) and moves towards the realaxis as the quark mass increases. For small densities we see that the curves turn aroundand tend towards smaller real values for larger quark mass. This behavior disappears whenwe raise the density up to a critical density dc ≈ 0.04.

6.2 Scalar

In order to compute the spectral function for scalar modes we first calculate the pullback ofthe metric to the D7-branes and expand in the fluctuations of the embedding variables δΘand δφ. δΘ corresponds to the scalar and δφ to the pseudoscalar excitations. We considertime and ρ dependence of the fluctuations only, since we stay at zero momentum. Theinduced metric then reads

ds2 =12ρ2

R2

(−f

2

fdt2 + fdx2

i

)+R2

ρ2

1− χ2 + ρ2χ′2

1− χ2dρ2 +R2 sin(Θ + δΘ)dΩ3

−2R2 χ′√1− χ2

∂aδΘdxadρ+R2∂aδΘ∂bδΘdxadxb . (6.7)

– 28 –

PSfrag repla ements11:52 2 2:2 2:4 2:6RewImw ~d = 0~d = 0:03~d = 0:04~d = 0:05~d = 0:1

(a)

2.05 2.10 2.15

-2.0

-1.5

-1.0

-0.5

(b)

Reω

d = 0

Imω

Figure 19: (a) First quasinormal modes for several densities d. For smaller densities a turningpoint in the real part occurs, which vanishes as the densities rises. (b) The first quasinormal modefor vanishing density. The arrows display the residue of the pole at the specific point, particularlythe direction its phase and the length its absolute value.

This coincides with the result found in [14] at zero density.To obtain a consistent solution at non-zero density, it is necessary to also include

fluctuations in the gauge field δA0 since these couple to the fluctuations δΘ of the inducedmetric of the D7-brane [24]. The coupling occurs since both fields transform as scalars underthe group of rotations SO(3). They cannot be distinguished by the different transformationunder the U(1) gauge symmetry anymore since this symmetry is broken at finite density.We may also think of the embedding scalar as being effectively charged and this explains itscoupling to the gauge field fluctuations. Then the action in this case differs from the actionin [14] by the non-vanishing gauge field and gauge field fluctuation terms. The action canbe found in appendix D. The equation of motion is given by

0 = ∂ρ[A∂ρ(δΘ)] +Bω2(δΘ) + C(δΘ) (6.8)

where

A =ρ5ff(1− χ2)3

(1− χ2 + ρ2χ′2)3/2

√1− 8d2

ρ6f3(1−χ2)3+8d2

,

B =ρf2(1− χ2)2√

1− χ2 + ρ2χ′2√

1− 8d2

ρ6f3(1−χ2)3+8d2

,

C =3ρ3ff(1− χ2)

√1− 8d2

ρ6f3(1−χ2)3+8d2√1− χ2 + ρ2χ′2

[1− 6χ

(ρf

fχ′ + χ

)],

(6.9)

with f and f defined in equation (3.2). For a more detailed derivation of the Lagrangianwe refer to appendix D.

– 29 –

0.95 0.96 0.97 0.98 0.99-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

χ0

Im w

Figure 20: The scalar quasinormal modes with pure imaginary quasinormal frequencies for dif-ferent densities d. For densities smaller than the critical density dc = 0.00315, the mode becomestachyonic in some region of the parameter space. The several colors corresponds to different den-sities d = 0 (black), 0.001 (blue), 0.002 (red), 0.0025 green, 0.003 (orange), 0.00315 (cyan), 0.0035(purple).

Now we can, similar to the procedure with the transversal vector modes, split the solu-tion in a regulating and a regular part, and compute the asymptotic solution for the latter.This allows us then to apply the shooting technique (see A) to compute the quasinormalmodes in the same manner as above.

Results for scalars First we study the density dependence of the mode with purelyimaginary quasinormal frequencies found at zero density in fig. 7. Our numerical resultsare shown in fig. 20. In this figure we observe that the critical parameter χcrit

0 , at whichthe instability occurs, i. e. the quasinormal frequency enters the upper half plane, increaseswith the density. Thus we write χcrit

0 (d). Also we note that at finite densities the modesbecome stable again at some larger value for χ0, which we denote by χcrit

0,2 (d). If we increasethe density further, we obtain a critical density dc = 0.00315 at which the mode is alwaysstable. Therefore the system is unstable in the parameter region χcrit

0 (d) < χ0 < χcrit0,2 (d)

and d < dc. The numerical values can be obtained from fig. 20. In section 6.5 as wellas in the introduction, we further discuss this mode and relate the instability found to aninstability already known in thermodynamics.

Let us now consider the behavior of the first quasinormal mode if we vary the densityd. In fig. 21 we present our numerical results. In this figure we observe that for eachfinite density we find that for large enough χ0 the quasinormal frequencies behaves in asimilar way. The real part increases while the imaginary part decreases as we increaseχ0. For smaller values of χ0 we observe three distinct movements of the quasinormalfrequencies. For small densities d < 0.1 the quasinormal frequency first follows the lineof the quasinormal mode corresponding to zero density as we increase the parameter χ0.After a critical value of χ0 is reached the quasinormal frequency leaves this line as the realpart of the quasinormal frequency increases monotonically. For slightly larger densities0.11 < d < 0.2, the frequencies also first moves along the line of the quasinormal mode atzero densities but in contrast to the case discussed above the real part of the frequencies

– 30 –

2 3 4 5 6 7-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Re w

Im w

(a)

1.8 2.0 2.2 2.4 2.6 2.8 3.0-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Re w

Im w

(b)

2 3 4 5 6 7-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Re w

Im w

(c)

Figure 21: Dependence of the first scalar quasinormal frequency on the density: In (a) and (b) thequasinormal frequencies for small densities d ≤ 0.2 are shown while in (c) densities up to d = 5 areconsidered. In (a) and (b) the different colors corresponds to distinct densities d = 0 (black), 0.01(green), 0.05 (blue), 0.1 (red), 0.11 (orange), 0.125 (brown), 0.15 (magenta), 0.2 (cyan). In (c) thecolor coding is d = 0 (black), 0.2 (green), 0.25 (blue), 0.5 (red), 1 (orange), 2 (brown), 5 (magenta).In all three plots the black dot marks the critical value of of the quark mass/temperature parameterχ0 where the instability occurs.

now first decreases. For even higher densities d > 0.2, the frequency at χ0 = 0 stronglydepends on the densities. We find from fig. 21 (c) that both the real and imaginary part ofthe quasinormal frequencies at χ0 = 0 increase with d. As we increase χ0 we find the usualbehavior, the real part of the frequency increases while the imaginary part decreases.

6.3 Schrodinger potential analysis

In this section we present the Schrodinger potential analysis which we introduced in section4 at finite baryon density. We use this analysis to explain the qualitative movement of thequasinormal frequencies as we change the baryon density.

For the vector fluctuations we observe in fig. 19 (a) a turning point in the movementof the quasinormal frequencies at low baryon density. The real part of the quasinormalfrequencies first increases and later decreases as we increase the mass parameter m. Thisturning point disappears as the critical density of dc = 0.04 is reached. After a criticalquark mass is reached the real part of the quasinormal frequencies always increases whilethe imaginary part decreases.

– 31 –

Let us now consider the Schrodinger potential which correspond to these quasinormalmodes in fig. 22. A similar analysis was also done in [23]. In addition to the infinitewall at R∗ = 0 which corresponds to the AdS boundary an additional peak appears in thepotential as we increase the quark mass parametrized by χ0. For small densities (see fig. 22(a)) this peak slowly grows out of the step-shape potential already observed in section 4.The step-shape potential is also present at zero density and it is known from the analysiswe presented there that the corresponding quasinormal frequencies show the turning pointbehavior discussed above. Thus the new feature of the finite density setup is the appearanceof the peak at high quark masses. As the peak grows, ‘bound’ states with positive energycan be formed, i. e. the real part of the corresponding quasinormal frequency is alwaysbigger than its imaginary part. Therefore we find quasiparticle excitations whose massesincrease as we increase the quark mass.

If we increase the baryon density, the peak already appears at lower quark mass andcan thus destroy the step-shape potential (see fig. 22 (b) and (c)). For instance a step isstill observable in the orange and blue curve in fig. 22 (b) while in fig. 22 (c) this stepis gone. Since we know that the step-shape potential is the reason for the turning pointpotential, we also find in this analysis a critical baryon density at which the turning pointdisappears. This critical density agrees with the value found in fig. 19 (a). For even largerdensities (see fig. 22 (d)) the peak increases even faster, i. e. the real (imaginary) part ofthe corresponding quasinormal frequencies increase (decrease) even faster.

Let us now consider the scalar fluctuations at finite density which show several distinctfeatures: a tachyonic mode at small densities and three qualitative different movements ofthe quasinormal frequencies.

From fig. 20 we know that there exists a parameter region where a scalar mode becomestachyonic. Especially interesting is that at finite density the mode is stabilized as the quarkmass is increased and that there is a critical density d = 0.00315 at which the mode isalways stable. In fig. 23 we present the Schrodinger potentials of the scalar fluctuations inthe relevant density region. As in the case of zero density a negative well appears in thepotential. It also first grows as we increase the quark mass (see fig. 23 (a)). However aswe increase the quark mass further, the well decreases while a peak forms in the potential.After a critical quark mass is reached, this well cannot longer support a ‘bound’ state withnegative energy, i. e. the imaginary part of the quasinormal frequency becomes negativeagain. At the critical density d = 0.00315 (see fig. 23), the potential also shows a negativewell. However this well just reaches a critical depth in order to support a zero energy‘bound’ state which is due to the zero-point energy (cf. the three dimensional potentialpot know from quantum mechanics).

Now we investigate the Schrodinger potentials which are relevant for the movement ofthe first quasinormal frequency studied in fig. 21. These potentials are plotted in fig. 24.The first observation is that for small densities d . 0.15 (see fig. 24 (a) and (b)) thepotentials at low quark mass agree with the potential at zero density. Thus also thecorresponding quasinormal frequencies must agree which we already found in fig. 21. Atlarger densities (see fig. 24 (c)) even the potential at zero quark mass differs from theone at the zero density such that the corresponding quasinormal frequencies at zero quark

– 32 –

0 1 2 3 4

R*0

1

2

3

4

5

PSfrag repla ementsVS

R(a)

0 1 2 3 4

R*0

2

4

6

8

10

PSfrag repla ementsVS

R(b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

R*0

5

10

15

20

PSfrag repla ementsVS

R(c)

0.0 0.5 1.0 1.5 2.0

R*0

5

10

15

20

25

30

35

PSfrag repla ementsVS

R(d)

Figure 22: Schrodinger potential of the vector fluctuations at finite baryon density (a) d = 0.003,(b) d = 0.01, (c) d = 0.03 and (d) d = 0.1. The different colors corresponds to distinct quarkmasses parametrized by χ0 = 0 (black), 0.5 (green), 0.9 (blue), 0.95 (orange), 0.97 (purple), 0.99(red). See figure 3 for the relation between χ0, the temperature and the quark mass Mq.

1 2 3 4

R*

-1

0

1

2

3

4

5

PSfrag repla ementsVS

R(a)

1 2 3 4

R*

-1

0

1

2

3

4

5

PSfrag repla ementsVS

R(b)

Figure 23: Schrodinger potential of the scalar fluctuations at finite baryon density (a) d = 0.002,(b) d = 0.00315. The different colors corresponds to distinct quark masses parametrized by χ0 = 0(black), 0.5 (green), 0.9 (blue), 0.95 (orange), 0.97 (purple), 0.99 (red). See figure 3 for the relationbetween χ0, the temperature and the quark mass Mq.

mass depend on the density which is consistent with the result found in fig. 21 (c). As forthe vector fluctuations a peak appears in the potential at finite density as we increase thequark mass. These peak can again support ‘bound’ states which correspond to quasiparticleexcitations. Thus the imaginary part of the quasinormal frequencies decreases while thereal part increases as we increase the quark mass. This is the overall movement of thequasinormal frequencies which is shown in fig. 21.

– 33 –

1 2 3 4

R*0

2

4

6

8

10

PSfrag repla ementsVS

R(a)

0.5 1.0 1.5 2.0

R*0

20

40

60

80

100

PSfrag repla ementsVS

R(b)

0.5 1.0 1.5 2.0

R*0

20

40

60

80

100

PSfrag repla ementsVS

R(c)

Figure 24: Schrodinger potential of the scalar fluctuations at finite baryon density (a) d = 0.01,(b) d = 0.15, (c) d = 0.5. The different colors corresponds to distinct quark masses parametrizedby χ0 = 0 (black), 0.5 (green), 0.9 (blue), 0.95 (orange), 0.97 (purple), 0.99 (red). See figure 3 forthe relation between χ0, the temperature and the quark mass Mq.

In general we observe that at finite baryon density a peak appears in the Schrodingerpotential as we increase the quark mass. If the peak is high enough, it separates thehorizon of the black hole from the AdS boundary. Only a small leak into the black hole ispossible due to tunneling. The potential thus approaches the one of a Minkowski embeddingwhere the brane does not fall into the horizon of the black hole and therefore can supportstable normal modes which are calculated in [8] in the supersymmetric limit. In fig. 25 weexplicitly confirm that the Schrodinger potential obtained from the black hole embeddingsconverges to the potential of the supersymmetric Minkowski embedding7 with the samemass parameter m as the mass parameter goes to infinity. This convergence explains thatalso the quasinormal frequencies approach this supersymmetric mass spectrum if the quarkmass is big enough. This behavior was already found for the vector fluctuations in [23,30].We can also understand this convergence if we look at the phase diagram (see fig. 26).For large quark mass over temperature ratios the equal density lines approach the shadedregion where the Minkowski embeddings are preferred. Since the phase transition is thirdor second order for large quark mass over temperature ratios as it is shown in [21], we alsoexpect a smooth transition from the spectrum of the quasinormal modes obtained from the

7The Schrodinger potential of a supersymmetric Minkowski embedding is given by [25]

VS = m2h1/4 + 3/8

“tan2(mR ∗ /

√2) + cot2(mR ∗ /

√2)”i

.

– 34 –

0.0 0.1 0.2 0.3 0.4 0.5

R*0

2000

4000

6000

8000

10 000

PSfrag repla ementsVS

RFigure 25: Comparison of the Schrodinger potential obtained from black hole embeddings (solidlines) and from supersymmetric Minkowski embeddings (dashed lines) at the same quark mass m.The different colors correspond to different quark masses parametrized by χ0 = 0.99 (black), 0.999(green), 0.9999 (blue) at a fixed density d = 0.5. See figure 3 for the relation between χ0, thetemperature and the quark mass Mq.

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1PSfrag repla ements M q M qTMTM~d = 0

~d = 4~d = 0:25~d = 0:00315

Figure 26: Sketch of the phase diagram: The chemical potential µ divided by the quark mass Mq

is plotted versus the temperature T divided by M = 2Mq/√λ. Two different regions are displayed:

The shaded region with vanishing baryon density where Minkowski embeddings are preferred andthe region above the transition line with finite baryon density where the black hole embeddingsare preferred. Here we work in the second phase. The curves are lines of equal baryon densityparametrized by d.

black hole embeddings to the spectrum of the normal modes obtained from the Minkowskiembeddings.

Next we would like to understand the appearance of the peak in terms of the D7-brane embedding. In [16] it was shown that the finite baryon density on the brane isinduced by fundamental strings which are stretched from the horizon of the black hole tothe D7-brane. At large quark masses these strings form a throat close to the horizon. Weconfirm numerically that the end of this throat and the peak in the Schrodinger potentialare located at the same value of the radial coordinate ρ (see fig. 27). Thus we can interpretthis throat as a potential barrier for the fluctuations which becomes bigger as the throatbecomes deeper.

6.4 Analytic solution at high frequencies

Motivated by the numerical solution to the fluctuation equations of motion shown in fig-

– 35 –

2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

PSfrag repla ements

(a)

2 4 6 8 10

0

100

200

300

400

PSfrag repla ementsVS

(b)

Figure 27: Comparison of the location of the peak in the Schrodinger potential and the throat inthe embedding of the D7-brane. The end of the throat is located where χ first changes its value.The different colors corresponds to different quark masses parametrized by χ0 = 0 (black), 0.5(green), 0.9 (blue), 0.95 (orange), 0.97 (purple), 0.99 (red). See figure 3 for the relation betweenχ0, the temperature and the quark mass Mq. The baryon density is d = 0.25.

PSfrag repla ements

sImF

ReF

00 1 2 3 4 5 6 70:60:40:2

0:20:40:6

PSfrag repla ements

sImF

ReF

00 1 2 3 4 5 6 70:60:40:2

0:20:40:6

Figure 28: Imaginary part of the solution to the regular function F versus the proper radialcoordinate s [38]. The left plot at vanishing baryon density d = 0 shows that the initially sinusoidalsolution is deformed as the mass parameter χ0 = 0.01, 0.5, 0.8, 0.9 is increased (see figure 3 forthe relation between χ0, the temperature and the quark mass Mq). Furthermore, its amplitudedecreases while the wave length increases. The right plot shows that introducing a finite baryondensity d = 0.2 causes the solutions to change their behavior with increasing χ0: While the firstthree curves for χ0 = 0.01, 0.5, 0.8 show the same qualitative behavior as those in the left plot, theblue curve for χ0 = 0.9 clearly signals a qualitative change with its increased amplitude. Lookingat the wave lengths in the lower plot we realize that already the green curve (χ0 = 0.8) shows adecreased wave length as well as the blue curve (χ0 = 0.9). The awkward oscillation pattern nears = 0 stems from a second mode being superimposed on the one we are tracking here.

ure 28, we suspect that this damped oscillating curve near the horizon can be approximatedby a damped quasi-harmonic oscillator, i.e. we should be able to find an approximate equa-tion of motion which is a generalization of the damped harmonic oscillator equation. Byquasi-harmonic we mean that the oscillator is damped with the damping depending onthe location of the mode in radial direction. From the observations in figure 29 we have

– 36 –

PSfrag repla ementssImFImF

00

1

1 2 3 4 5 6 710:50:5

PSfrag repla ements

s

ImF

ImF 00

1

1 2 3 4 5 6

710:5

0:40:20:20:40:60:8

0:5Figure 29: Left: Solutions to the damped harmonic oscillator equation (6.14) for increasingdamping coefficient γ = 0, 0.1, 0.2, 0.3, 0.9 (black, red, green, blue, orange). Here the eigenfre-quency ω is fixed to 1 in analogy to the plots in figure 28. Right: Hi-frequency (ω = 50) numericalsolution showing that the damping of the mode is strong near the horizon s = 0 but decreasesexponentially in the bulk towards the boundary s→∞.

already concluded that the amplitude changes rapidly near the horizon and change less inthe bulk until the boundary is reached. Thus it is reasonable to assume that the dampingof the mode F already is strong near the horizon and a near-horizon approximation cancapture this effect to certain extent. In this spirit we take the near-horizon limit % ∼ 1 andat the same time the high-frequency limit ω 1.

Applying these limits for the flat embedding χ0 = 0 in the equation of motion (6.5),we obtain the simplified equation of motion

yH ′′ + (−iω − y)H ′ + iω

2

(1√7

+ 1)H = 0 , (6.10)

where the variable is y = iω√

74 (% − 1) and the regular function H(y) comes from the

Ansatz E = (% − 1)βF with the redefinition F = e−√

7/4iω(%−1)H. This equation of mo-tion has the form of Kummers equation, which is solved by the confluent hypergeometricfunction of first H = 1F1[−iω(1/

√7 + 1)/2,−iω, %− 1] and second kind U . Boundary con-

ditions rule out the second kind solution which is non-regular at the horizon and thereforecontradicts the assumptions put into the Ansatz E = (% − 1)βF . Since we are interestedin how the solution changes with decreasing m, we need to choose χ0 non-vanishing. Alsowith this complication we still get Kummers equation with changed parameters and theanalytic solution for F is given by

F = e−iω

2(%−1)

r74

+4χ2[χ0,d]

2

1−χ20 1F1

[− iω

2

1

2√

74 + 4χ2[χ0,d]2

1−χ20

+ 1

, −iω,

iω(%− 1)

√74

+4χ2[χ0, d]2

1− χ20

],

(6.11)

– 37 –

PSfrag repla ements

0:1 0:2

0:5

0:1750:150:1250:0750:050:02501

0:51ImF

Figure 30: Approximate analytic solution compared to the exact solution at ω = 70 , d = 0 , χ0 =0.4 (see figure 3 for the relation between χ0, the temperature and the quark mass Mq).

with the near horizon expansion of the embedding function χ = χ0 +χ2[χ0, d](%−1)2 + . . .

where we determine recursively

χ2[χ0, d] = 3χ0χ6

0 − 3χ40 + 3χ2

0 − 14(1− 3χ2

0 + 3χ40 − χ6

0 + d2). (6.12)

The approximate solution for F is shown in figure 30. Furthermore we can calculate thefraction ∂4E/E appearing in the spectral function near the horizon using this analyticsolution. The result is displayed in figure 31. This near horizon limit is not the spectralfunction since we would have to evaluate it at the boundary which lies far beyond thevalidity of the near horizon approximation. Nevertheless, in the spirit of [39] and accordingto our initial assumptions that the effect of damping mainly takes place near the horizonwe further assume that the limit shown in figure 31 already contains the essential featuresof the spectral function. Indeed the fraction shows distinct resonance peaks which moveto lower frequencies if we increase the mass parameter m. The right picture shows thesame situation at a finite baryon density d = 1 and we see that the peaks do not move tolower frequencies as much as before. Thus also the vanishing of the turning point at largedensities as observed before is captured by this approximate solution.

6.5 Discussion: Turning point and Tachyon

We now discuss the turning point in the spectrum of the vector modes and its dependenceon the density. For scalars the purely imaginary scalar quasinormal mode reaches intothe upper half of the complex plane and thus yields a tachyonic excitation, which we alsodiscuss here.

– 38 –

PSfrag repla ements E E

w0 0 0:2 0:4 0:6 0:8 1 1:2100200300400500

PSfrag repla ements E E

w0 0 0:2 0:4 0:6 0:8 1 1:2100200300400500

Figure 31: Approximate spectral function fraction near the horizon computed with the func-tion E = (ρ− 1)βF (ρ) and F being the analytic approximation given in equation (6.11).

6.5.1 Turning point

In this section we discuss why the vector mesons at finite temperature get smaller massesas the quark mass is increased. Further we investigate why there is a turning behavior atfinite density. In the latter case we also find an analytic solution valid at high frequencies.

Line of argument Let us first summarize our results. The D3/D7-system at finitedensity has two competing geometrical features. One is the formation of a potential barriernear the horizon due to the charge located at the horizon. This causes the resonancesin the spectral function to become more stable and to move to larger frequency whenthe quark mass is increased. The second feature is the length of the brane 8 supportingan excitation from the horizon to the boundary. Increasing this length causes the vectormeson resonances to move to smaller frequency with increasing quark mass. These twogeometrical features compete at finite density while at zero density the potential barrier isabsent. Below we will explain this in greater detail and give a field theoretic interpretation.

Guiding features of numerical solutions A detailed analysis of the numerical solu-tions [38] shows some interesting features. Most prominently there is a turning behavior, i.e.the location of the vector resonance peaks first move towards lower real frequencies, thenturn around and move towards higher ones in order to asymptote to the SUSY-spectrum(increasing the quark mass to infinity is equivalent to the zero temperature case). Thisturning point for the lowest of the vector resonance peaks is only present at small densitiesd . 1. For larger densities the peaks run towards higher frequencies with increasing quarkmass from the start, i.e. there is no turning point. As we see from the vector quasinormalmodes the imaginary part monotonically decreases with increasing quark mass. At vanish-ing density the resonances only move to lower frequencies with increasing quark mass inthe regime of black hole embeddings.

Therefore we have two things to explain: 1) the turning of the real part of the frequencyand the decreasing imaginary part and 2) the motion towards smaller frequencies at zerodensity or small densities and quark masses (d < 1, χ0 < 0.5). The solutions E of the bulk

8Since the AdS-boundary is infinitely far away from the horizon, the length of the brane has to be

renormalized by subtracting for example the length of the brane at trivial embedding χ0 = 0.

– 39 –

I II IIIPSfrag repla ementsRbdy Rbarrier RAx

Figure 32: Schematic outline of the solution Ax(R∗) to the vector fluctuation equation of mo-tion (6.5). The horizon is located at R∗ =∞ where the wave function diverges, the AdS-boundaryis R∗bdy = 0. The solid black curve shows the solution at finite temperature, density and quarkmass, i.e. on a black hole embedding. The dashed blue curve shows the corresponding solution ona Minkowski embedding at the same quark mass. The three regimes correspond to the nature ofthe potential shown in figure 25: I) Minkowski-like, II) potential barrier, III) essentially vanishingpotential.

fluctuation equation of motion are composed of a singular part and the regular part F .From these solutions the spectral functions are computed. The regular part of the solutionshas interesting properties, in particular:

1. The proper distance s on the brane

s =

ρ∫ρH

√Gρρdρ , (6.13)

increases with increasing quark mass parameter χ0. Integrating this expression fromthe horizon, to the boundary gives the proper length of the D7-brane. This alsochanges the wavelength of the fluctuations on the brane, as we will explain below.

2. As we see from figure 28 the form of the numerical solutions written in the properradial coordinate s shows a damped oscillation. Thus we are motivated to map thefluctuation equation of motion to a damped harmonic oscillator equation of motion.

3. At vanishing density d = 0 a larger quark mass parameter χ0 induces a strongerdamping of the solutions F (s) (see figure 28, left).

– 40 –

4. At finite density a larger quark mass parameter χ0 first induces a stronger, then aweaker damping of the solutions F (s) (see figure 28, right).

5. As seen from the high frequency (ω = 50) solution in figure 29 most of the dampingoccurs close to the horizon while the fluctuation propagates towards the boundarywith exponential damping.

6. At d = 0 for χ→ 1 black hole embeddings asymptote tom = ρχ|ρ→ρbdy ≈ 1.3; i.e. thisgeometric construction can not produce the T → 0 limit in the black hole embeddings.Furthermore, long before χ0 → 1 one has to jump to Minkowski embeddings at thetemperature where the instability appears. In other words: at fixed T you can notmake the quarks heavier than Mq ∼ 1.3T . And there is no smooth transition throughthe singularity at the limiting embedding χ0 = 1 to the Minkowski embeddings.

Resonances moving to higher frequencies towards the SUSY spectrum At finitedensity there are two effects present which compete. One effect is the elongation of thebrane. This is discussed below for the case of vanishing density. A specialty at finitedensity is the appearance of a longer and longer spike in the embedding reaching down tothe horizon, connecting the Minkowski-like bulk part of the black hole embeddings withthe horizon.

The second and as it turns out also the more important effect at finite density is theformation of a potential barrier near the horizon at R∗ = ∞, see figure 25. This barriereffectively cuts off the horizon from the geometry. Only a small part of the wave function”leaks” into the region behind the potential barrier where the black hole is located, that isregion III in the schematic solution shown in figure 32. In this region III the Schrodingerpotential asymptotically vanishes. Therefore the solution shown in figure 32 first dropsto low values in order to diverge only very close to the horizon (not shown in the figure).Region II is the finite radial distance covered by the potential barrier. Here the wavefunction qualitatively drops exponentially. In region I the main part of the solution islocated between the boundary and the barrier. Here the potential approaches a pot-likeform more and more similar to the corresponding potential generated by the Minkowskiembedding as the quark mass parameter is increased. As this happens, the black holefluctuation solution (solid black curve in the schematic plot in figure 32) approach theMinkowski solutions (dashed blue curve in the schematic plot in figure 32).

As we see from figure 27 the potential barrier moves closer to the boundary as the massparameter χ0 is increased. Thus the potential gets more narrow and the lowest possibleexcitation is raised to a higher energy, i.e. the real part of the corresponding quasinormalmode is increasing. Furthermore the barrier becomes higher such that the correspondingexcitation becomes more stable, i.e. the imaginary part of the quasinormal frequencydecreases. In other words, less of the wave function leaks over the barrier into the blackhole.

Resonances moving to lower frequencies Let us for simplicity work at vanishingdensity first. All our considerations will also apply to the finite density case. The heuristic

– 41 –

explanation for the left-movement of resonances has to do with the proper length of theD7-brane. At χ0 = 0 we have a flat embedding which is the minimal length the branecan have when measured in s. Increasing the quark mass parameter χ0 the D7-brane be-comes ”longer” in the sense that the proper length computed in the coordinate s increases.Therefore we can imagine a solution supported on the brane to be ”stretched” togetherwith the brane, i.e. its effective wavelength increases. Assuming a constant effective speedof light, the effective frequency of this solution has to decrease. In this sense χ0 here actsanalogously to a damping coefficient γ in a damped harmonic oscillator. 9

We may quantify this intuition examining the solutions F (s) shown in figure 28. In-deed, comparing to the solutions (figure 29) of the damped harmonic oscillator equation

0 = ∂2tX(t) + 2ω0γ∂tX(t) + ω0

2X(t) , (6.14)

with damping coefficient γ, we find that increasing χ0 in figure 28 resembles the effectof increasing the damping coefficient γ in the analogous harmonic oscillator solutions,figure 29.

This leads us to assume that the fluctuation equation of motion on the D7-branefor real values of the frequency ω ∈ R can effectively be replaced by the equation ofmotion for a damped harmonic oscillator with an effective eigenfrequency ωeff and withthe replacements γ → γ(χ0), with γ(χ0) a monotonous function and ω0 → ω(χ0 = 0).By ω(χ0 = 0) we mean the effective eigenfrequency corresponding to the solution mainlyinfluenced by the lowest of the quasinormal modes. γ(χ0) can be seen as the effectivedamping coefficient 10. So the effective solution at finite χ0 reads

F (s) ∝ e−γ(χ0)ω0seisωeff (χ0) (6.15)

with the reduced eigenfrequency

ωeff (χ0) = ω0

√1− γ(χ0)2 . (6.16)

Since 0 ≤ γ(χ0) ≤ 1 and γ(χ0) is monotonous, the frequency of the solution which hasthe eigenfrequency ω0 at χ0 is monotonously decreasing with increasing χ0. So the quarkmass parameter χ0 effectively acts as a damping coefficient. We might even suspect thatthe embedding χ(ρ) acts as a local damping coefficient depending on the radial location.In this way the fact that the damping is strongest at the horizon and vanishes towards theboundary (see figure 29, right) is consistent with the fact that the embedding χ(ρ) assumesits largest value near the horizon and quickly asymptotes to zero towards the boundary.

9This heuristic picture neglects the fact that the damping of the gravity modes in general is a local

effect, i.e. it depends on the radial AdS-coordinate (through radially-dependent geometry). Tentatively we

will assume that we can average the damping effects over the radial coordinate and express them globally

in an effective damping coefficient independent from the radial coordinate.10This identification of γ(χ0) is effective in the sense that the actual quantity appearing in the equation

of motion (6.5) is χ(ρ) which highly depends on the radial location. So we understand the effective damping

γ(χ0) to be a constant which averages damping effects over the whole AdS-radius.

– 42 –

Now let us (as a very crude approximation) choose ω0 to be the real part of the firstvector quasinormal mode on the D7-brane at vanishing χ0. The quasinormal mode is al-ready damped, i.e. it actually has no real eigenfrequency. But let us nevertheless follow ourrecipe and replace the complicated fluctuation equation by the simple damped harmonicoscillator with an effective damping γ(χ0) and effective reduced eigenfrequency ω(χ0) fromequation (6.16). The decreasing eigenfrequency ω(χ0) with increasing χ0 effectively ex-plains the left-motion of the resonances in the corresponding spectral functions computedfrom these solutions ImGR ∝ ∂sF (s)/F (s) |s→sbdy . Here we have assumed that the lowestresonance peak in the spectral function behaves in the same way (moving to lower fre-quencies) as the effective eigenfrequency ωeff (χ0) of the solution F (s) from which it iscomputed. This intuition we get from the fact that in the exact computation both of thesefrequencies are mainly determined by the behavior of the lowest quasinormal mode.

These considerations shall serve to give an intuition for the behavior of the resonances.To be more precise the resonances are actually influenced by all the quasinormal modes,i.e. by their location in the complex frequency plane and by their residues.

6.5.2 Killing the Tachyon

The scalar tachyon appearing at zero density is stabilized by introducing baryon charge.From a critical density dc = 0.00315 on the scalar does not become tachyonic for any valueof χ0. Thus the finite charge density d cures the instability and stabilizes the black holephase of the D3/D7 system 11. We have described the scalar quasinormal mode signaturesin detail in section 6.2. The mechanisms explaining this effect are discussed with the helpof Schrodinger potentials in section 6.3. The negative potential well supporting the tachyonis lifted with increasing charge density. As discussed before the appearance of the tachyonis at non-zero density connected to the black hole to black hole first order phase transitiontaking place at finite densities 0 < d ≤ 0.00315 between two distinct black hole phases.In particular the tachyon appears on the unstable branch of the free energy diagram ofthe phase transition. This branch connects two metastable branches as shown in figure 1.At the critical density dc = 0.00315 both the tachyon and the black hole to black holephase transition disappear. It is not clear what is the physical ground state below dc. Aswas argued in [17] the true ground state might be a mixed phase for which the gravitydescription is not known so far.

7. Conclusions and Outlook

Our extensive study of the quasinormal modes of the D3/D7-brane system has revealedsome interesting relations between previously known and unknown phenomena. As amain result we have shown that the system is completely stable above the critical den-sity dc = 0.00315. That means that the spectrum of scalar excitations does not containany tachyonic mode. Also in this regime the spectrum of mesonic exictations in the fieldtheory –corresponding directly to distinct quasinormal modes– is de-singularized at low

11Note that there exist Minkowski embeddings with the same chemical potential and the same quark

mass, but all states in the Minkowski phase have strictly vanishing density d = 0 [16,17].

– 43 –

temperature or large quark mass through the explicit breaking of a scaling symmetry nearthe limiting embedding. I.e. the different meson excitations behave in accordance withthe mass formula derived for the supersymmetric case [8]. Furthermore for the regimebelow the critical density, i.e. for d < 0.00315 we have established the connection betweenthe black hole to black hole phase transition at finite charge density on one hand and thetachyonic scalar on the other. Using the Schrodinger formulation of the problem we haveexplained the movement of scalar and vector quasinormal modes in the complex frequencyplane in great detail. A universal feature of all Schrodinger potentials at finite charge den-sity is that they develop a barrier near the black hole horizon which hides the horizon andthe black hole from the boundary. In consequence the dissipation decreases with increasingdensity and the quasinormal modes asymptote to a normal mode behavior. This behaviorhas also been observed in [40] in a setup where a black hole in AdS4 develops scalar hair,i.e. there is also a charge density distributed near the horizon.At vanishing baryon density but finite momentum we found a critical wavelength at whichthe hydrodynamic approximation explicitly breaks down. Below that wavelength the sys-tem at late times is no longer governed by hydrodynamic but by (propagating) collisionlessmodes only. A few unresolved issues remain for zero baryon density. In that case we founda spiraling behavior of the first quasinormal mode’s trajectory for both scalars and vec-tors. The number n of loops appears to be directly related to what we coined ”attractor”frequencies ωn to which all QNM-trajectories asymptote if their momentum kn lies in acertain momentum regime kn−1 < kn ≤ kn+1. A direct relation remains unrevealed. Inthe same case we found that a scalar becomes tachyonic at low enough temperature. Thisinstability might give rise to a condensation process or more generally to a phase transi-tion as suggested by the hydrodynamic behavior of the scalar mode where it just becomestachyonic. It is interesting to ask what the new phase could be and if it exists at all.The analysis presented here can straightforwardly be extended to the D3/D7 setup withfinite isospin density or with spontaneously broken symmetry [41,42]. In these setups thereis a variety of modes to be studied, in the latter most prominently the fluctuations of theorder parameter phase which correspond to Goldstone modes becoming new hydrodynamicmodes in the phase with spontaneously broken symmetry.

Acknowledgements

We are grateful to J. Mas, F. Rust, J. Shock, J. Tarrıo for discussions. The work ofK.L. M.K and F.P.B has been supported by Plan Nacional de Alta Energıas FPA-2006-05485, Comunidad de Madrid HEPHACOS P-ESP-00346, Proyecto Intramural de CSIC200840I257. F.P.B has also been supported by the Programa de apoyo institucional paracursar estudios de doctorado de la Universidad Simon Bolivar.The work of J.E. C.G. and P.K. was supported in part by The Cluster of Excellence for Fun-damental Physics - Origin and Structure of the Universe. The work of M.K. was supportedin part by the German Research Foundation Deutsche Forschungsgemeinschaft (DFG).

– 44 –

A. Shooting method

We now discuss an improved shooting method for solving the equation of motion for thefluctuations at complex frequencies. The problem arises when using the standard methodfor solving the equations of motion by just numerically integrating the differential equation.

As an example for the failure of this naive method we look at figure 33, which shows thespectral function for the transverse vector modes for vanishing quark mass and vanishingdensity. Note here that we use the dimensionless frequency w = ωph/(2πT ) = ω/2 andmomentum k = kph/(2πT ) = k/2. The line of poles found at Im w = −1 is definitelynot correct since in [43] the pole-structure of the very same configuration was analyticallydetermined to be

w = n (1− i) for n ∈ N0 . (A.1)

It turns out that the numerical errors hide the quasi-normal modes behind this ”wall“at Im w = −1. This wall could be misinterpreted as a branch cut.

The problem is, that the equations of motion we investigated are not regular at thehorizon where the initial conditions are imposed. One therefore splits the solution in aregular and regulating part: E(ρ) = (ρ− 1)−iw F (ρ) by computing its Frobenius expansionnear the horizon. Nevertheless, we have to move the starting point for the numericalintegration slightly away from the real horizon to, say ρ = 1.00001 where we choose thisvalue so close to 1 that we can expect only small deviations from the exact solution.However, for values of Im w ≤ −1 the boundary condition yields

(ρ− 1)−iw =(1× 10−5

)Im w−iRe w. (A.2)

This is problematic when Im w is smaller than −1, the initial condition is becominglarge, leading to round off errors. These change the value of the fluctuation at infinitydramatically and are responsible for the invalid results of the spectral function in thespecific region.

The cure is to shift the starting point for the numerical integration away from thehorizon, and thus dealing with not so small values in the basis of (A.2), resulting in betternumerical initial conditions. Of course, the error from starting the integration furtheraway from the horizon has to be compensated. This is simply done, by calculating theasymptotics of the gauge field fluctuations in the vector case, or the embedding deviationsin the scalar case at the horizon to higher order.

When reimplementing the numerical integration with a starting value of e. g. ρ = 1.1and the new asymptotics, the numerical aberration is prevented successfully and one cangain insight farther into the complex ω-plane.

A look at the results shows what can be achieved with this more sophisticated method.A comparison between the surface plot of the old method (figure 33) and the improved one(figure 34) reveals that the expected pole structure can now be seen clearly. In fact it canbe checked that the position of the poles agrees with the analytical result (A.1) very well.

– 45 –

Figure 33: Breakdown of the standard numerical techniques at Im w / −1. A series of spikes canbe seen, being a firmly erroneous solution in view of the analytic result (A.1). This problem willbe solved by means of our improved method, discussed in the text.

Figure 34: With the improved method nearly four poles can be resolved for vanishing densityd = 0 and massless quarks m = 0. For the computation of the initial values the expansion in thefluctuation equation were evaluated up to eleventh order. As can be seen, the location of the polesfits the analytical solution (A.1) very well.

B. Relaxation method

The determination of quasinormal frequencies can be understood as a particular case of atwo-point boundary value problem. A common numerical method for solving such problemsis the shooting method, where one solves the differential equations with varying boundaryconditions at one boundary and searches for a solution that approximates the wantedboundary condition at the other boundary within some numerical error. We describe thatmethod in appendix A.

A different approach is provided by the relaxation method which allows to fix thecorrect boundary values on both boundaries. Since this method is less frequently used wegive a short outline below12.

The method is based on replacing the differential equations by a system of finite-difference equations (FDE) on a discrete grid. Starting from an ansatz solution obeyingthe correct boundary conditions, one varies the value of the dependent variables at eachpoint relaxing to the configuration which provides an approximate solution for the FDE

12A more detailed exposition can be found in [44].

– 46 –

within some given numerical error. In our case, we convert our second order complexODE into a set of four first order equations by separating real and imaginary parts of thedependent variables.

More generally we can consider a set of N first order ODE’s

dyi(x)dx

= gi(x, y1, . . . , yN ;λ) , (B.1)

where each dependent variable yi(x) depends on the others and itself, on the independentvariable x and possibly on additional parameters, like λ above. In our case this (complex)parameter is the quasinormal frequency. These extra parameters can be embedded intothe problem by writing trivial differential equations for them yN+j ≡ λj ,

dyN+j

dx= 0 , since it is constant .

(B.2)

We assume that there are n additional parameters and include them from now on into theset of dependent variables y1, . . . , yN , N = N + n.

The solution to the problem involves N×M values, for the N dependent variables in agrid of M points. We also have to fix N boundary conditions for the dependent variables.

The system is discretized as usual

x→ 12

(xk + xk−1) , y → 12

(yk + yk−1) , (B.3)

for points in the bulk. One may arrange the whole set of yi’s in a column vector yk =(y1, . . . , yN )kT , where the subscript k refers to evaluation at the point xk, k = 1, . . . ,M.

With this matrix notation, the system (B.1) can be written as

0 = Ek ≡ yk − yk−1 − (xk − xk−1) gk(xk, xk−1,yk,yk−1) , k = 2, . . . ,M , (B.4)

where the Ek are the aforementioned FDE’s. These are the equations that we need to fulfil.Notice there are N equations at M − 1 points, so the remaining N equations are suppliedby the boundary conditions. We will set n1 of them on the left at x1, called E1, and therest n2 = N − n1 at xM , called EM+1.

Now we take a trial solution that nearly solves the FDE’s Ek. By shifting each solutionyk → yk + ∆yk and Taylor expanding in the shift, one obtains the relation

0 = Ek(y + ∆y) ' Ek(yk,yk−1) +N∑n=1

∂Ek

∂yn,k−1∆yn,k−1 +

N∑n=1

∂Ek

∂yn,k∆yn,k , (B.5)

⇒ −Ej,k =N∑n=1

(Sj,n ∆yn,k−1

)+

2N∑n=N+1

(Sj,n ∆yn−N,k

). (B.6)

This allows to find the ∆yk that improve the solution. First we merge the two differentials

Sj,n =∂Ej,k∂yn,k−1

, Sj,n+N =∂Ej,k∂yn,k

, n = 1, . . . , N , (B.7)

– 47 –

in a N × 2N matrix, for each bulk grid position xk. For the boundaries the expressionsfollow equally

−Ej,1 =N∑n=1

Sj,n ∆yn,1 =N∑n=1

∂Ej,1∂yn,1

∆yn,1 , j = n2 + 1, . . . , N , (B.8)

−Ej,M+1 =N∑n=1

Sj,n ∆yn,M =N∑n=1

∂Ej,M+1

∂yn,M∆yn,M , j = 1, . . . , n2 , (B.9)

where n runs in both from 1 to N . The whole (NM × NM) matrix S possesses a blockdiagonal structure. In fact since for reasonable systems N << M S is a sparse matrix.This allows the usage of computer packages in which the solution of linear systems withsparse matrices can be done in a very efficient way. The actual solution can now be foundby an iterative process until a desired accuracy is achieved. As measure for the discrepancyof an approximate solution to the actual solution we used

err =1

MN

M∑k=1

N∑j=1

∣∣∣∣ ∆y[j][k]scalevar[j]

∣∣∣∣ < conv , (B.10)

where scalevar[j] is an associated scale for each of the dependent variables (e.g. the valueat the midpoint or so). The idea is that when that averaged value of the shift to get abetter solution is smaller than conv, we accept the former values we had as the actualsolution. In our computations we set conv = 10−6.

In order to obtain the correct boundary conditions we have used the z coordinatesystem where the horizon lies at z = 1 and the boundary at z = 0. We split off the ingoingboundary condition on the horizon according to Φ(z) = (1 − z)−iω/4y(z) . Demandingy(1) = 1 and y(0) = 0 gives four real boundary conditions. Since in total we have howeversix dependent variables, counting also the real and imaginary part of the quasinormalfrequency we need two more boundary conditions. We found it convenient to expand thefunction y(z) in a Taylor series at the horizon and compute also y′(1) which provides theadditional two real boundary conditions. The typical gridsize we used was consisted of 5000points. Finally we note that we have implemented to outlined algorithm in GNU/Octave[45].

C. Schrodinger potentials

This appendix shows how to compute the effective potentials, i.e. the Schrodinger poten-tials for the scalar and vector fluctuations on the D7-probe-brane. In order to do so thelinearized fluctuation equations of motion have to be rewritten in terms of a new radialcoordinate R∗. This procedure has been described before (e.g. in [14], [25], . . . ), never-theless we include it here for completeness. For convenience we stick to the notation madeuse of in [14], and we compute all Schrodinger potentials in the ρ-coordinates introducedin section 3.

We are not interested in the higher angular excitations on the S3, so we separate thefluctuations according to φ(ρ, S3) = y(ρ)Φ(S3). Let us consider exclusively fluctuations

– 48 –

without angular momentum on the S3, i.e. Φ(S3) = 1. All the linearized vector and scalarfluctuation equations of motion are second order ordinary differential equations and canbe rewritten in the form

−H0

H1∂ρ [H1∂ρy(ρ)] +

[k2H2 +Hθ

]y(ρ) = w2y(ρ), (C.1)

where H0, H1, H2 and Hθ are in general functions of ρ and depend on the particular fieldfluctuation considered. Hθ only appears in the scalar fluctuations. For transverse vectorfluctuations we have

H0 = −gρρ

gtt, H1 =

√−ggρρgxx, (C.2)

H2 = −gzz

gtt, Hθ = 0 . (C.3)

For longitudinal vector fluctuations we have

H0 = −gρρ

gtt, H1 =

√−ggρρgzz, (C.4)

H2 = −gzz

gtt, Hθ = 0 . (C.5)

For scalar fluctuations the equations of motion do not take such a simple general formin terms of metric components since the metric itself contains scalar fluctuations. So wechoose to write explicitly

H0 = − f2ρ4(1− χ2)8f(1− χ2 + ρ2χ′2)

, H1 = ffρ5(1−χ2)3

(1−χ2+ρ2χ′2)3/2

√1 + 8d2

f3ρ6(1−χ2)3, (C.6)

Hθ =3f2f2ρ8(1−χ2)2[f3ρ6(1−χ2)3+8d2(1−6χ2)−144d2f3fρ9χχ′(1−χ2)2]

8[f3ρ6(1−χ2)3+8d2]2,

where we do not include H2 since the above coefficients are computed in the case of vanish-ing momentum k = 0 but at finite density d. At finite momentum and at vanishing densitywe have H2 = f2/f2 in addition to the d → 0 limits of H0, H1 and Hθ from (C.6). Atfinite momentum and density the scalar fluctuations couple to the vector fluctuations andwe will not address this complication in this work.

In any case we can substitute y(ρ) = hψ in equation (C.1) to obtain

−H0ψ′′ −H0

(2h′

h+H1′

H1

)ψ′ +

[k2H2 +Hθ −H0

(h′′

h+H1′

H1

h′

h

)]ψ = w2ψ . (C.7)

Introducing the new radial coordinate

R∗ =

∞∫ρ

dρ/√H0(ρ) , (C.8)

we can rewrite the first term −H0ψ′′ = −∂R∗2ψ + H0

′ψ′/2. The special choice of h =H0

1/4/H11/2 eliminates all the terms containing ψ′. Thus the fluctuation equation of

motion finally assumes Schrodinger form

−∂R∗2ψ + VSψ = Eψ (C.9)

– 49 –

with the effective energy E = w2 and the Schrodinger potential given by

VS = −H0

(h′′

h+H ′1H1

h′

h

)+ k2H2 +Hθ . (C.10)

From this general formula the specific scalar, longitudinal and transversal vector Schrodingerpotentials may be obtained by substituting for H0, H1, H2, Hθ, h using the correspondingvalues from equations (C.6), (C.4) and (C.2), respectively.

D. Finite density but vanishing momentum

To abbreviate the longish Lagrangian we introduced the following notation

a = 1− χ2, b = a+ ρ2χ′2, c =

8d2

ρ6f3a3 + 8d2, so 1− c =

ρ6f3a3

ρ6f3a3 + 8d2. (D.1)

In terms of these expressions the Lagrangian, expanded to second order, reads

L(−NfTD7

√h3

r4H4

) = L0 + L1 + L2 − ρ3ffa√b√

1− cF 04(δF40)− 32ρ3ff

a− bc√b√

1− c(δθ)2

− R4

r2H

ρf2

f

a2

√b√

1− c(∂tδθ)2 +

12ρ5ff

a2(a− bc)b3/2(1− c)3/2

(∂ρδθ)2

+R2

2

∑i

Gii0 χ2(∂iδφ)2 + ρ5f3/2χ′√

2rH

a2√cb (1− c)3/2

(δF40)(∂ρδθ)

− ρ3f3/2χ3√

2rH

a√c√

1− c(δF40)(δθ) +

14ρ3ffa

√b√

1− c∑ik

Gii0 Gkk0 (δFik)2

− ρ3ffa√b√

1− cF 04∑i

Gii0(−(1 + δ4i)R2χ′√

a

)(δFi0)(∂iδθ) (D.2)

where L0 is simply the Lagrangian without any fluctuations but nonvanishing density

L0 = ρ3ffa√b√

1− c (D.3)

and the boundary terms L1 and L2 are given by

L1 = ∂ρ

[−ρ

5ffa3/2χ′√b√

1− c(δθ)

]and L2 = ∂ρ

[−3

2ρ5ffχχ′a√b√

1− c(δθ)2

]. (D.4)

The symbols denoted by G with upper indices are the components (diagonal part) of theinverse background tensor G = G+ 2πα′F . The most important ones are

G000 = −2

R2

r2H

f

f2ρ2(1− c)and G44

0 =ρ2a

R2b(1− c). (D.5)

– 50 –

For the off-diagonal part of the inverse background tensor G04 ≡ F 04 = −F 40 holds, becauseof the antisymmetry of the field strength tensor F , and has the value

F 04 = −

√2fac

rHf√b(1− c)

. (D.6)

Finally, δFij = ∂iδAj − ∂jδAi is the field strength for the vector fluctuation.

E. Results for second quasinormal modes

In this appendix we collect results for the location of the second quasinormal modes (scalar,transverse and longitudinal vectors) at finite momentum but vanishing density.

E.1 Transverse vector fluctuations

Figure 36 shows the dispersion relation of the second transverse vector QNM at differentvalues for the mass parameter θ0.

-- -- -- --

2 3 4 5 6ΩR

1

2

3

4

-ImHΩL

k=4

k=3

k=2

k=1

--

--

3 4 5 6 7ReHΩL

0.5

1.0

1.5

2.0

2.5

3.0

3.5

-ImHΩL

k=5k=4.8k=4.4k=4.16k=4.12k=4.08k=4

Figure 35: Second quasinormal mode of transverse vector fluctuations for distinct values of thespatial momentum k.

1 2 3 4 5k

4.5

5.0

5.5

6.0

6.5

7.0

ImHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

1 2 3 4 5k

-3.8

-3.6

-3.4

-3.2

-3.0

-2.8

-2.6

ImHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

Figure 36: Dispersion relation for the second transverse vector quasinormal mode.

– 51 –

E.2 Longitudinal vector fluctuations

-- -- --

2.5 3.0 3.5 4.0 4.5 5.0ReHΩL

1

2

3

4

-ImHΩL

k=2.8

k=2

k=1

----

2.5 3.0 3.5 4.0 4.5 5.0ReHΩL

0.5

1.0

1.5

2.0

2.5

3.0

-ImHΩL

k=3.4

k=3.2

k=3

----

--

3 4 5 6ReHΩL

0.5

1.0

1.5

2.0

2.5

3.0

-ImHΩL

k=5k=4k=3.8k=3.4

Figure 37: Second quasinormal mode for longitudinal vector fluctuations at distinct spatialmomenta k.

1 2 3 4 5k

4.5

5.0

5.5

6.0

ReHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

1 2 3 4 5k

-3.5

-3.0

-2.5

ImHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

Figure 38: Dispersion relation for the second longitudinal vector quasinormal mode fluctuationsat distinct spatial momenta k.

– 52 –

E.3 Scalar fluctuations

---- ----

--

2 3 4 5 6 7ReHΩL

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-ImHΩL

k=5k=4k=3k=2k=1

Figure 39: Second scalar quasinormal mode at distinct k.

1 2 3 4 5k

5.0

5.5

6.0

6.5

7.0

ImHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

1 2 3 4 5k

-3.6

-3.4

-3.2

-3.0

ImHΩL

Χ0 = 0.72

Χ0 = 0.61

Χ0 = 0.48

Χ0 = 0.34

Χ0 = 0.20

Χ0 = 0

Figure 40: Dispersion relation for the second scalar quasinormal mode for distinct mass ortemperature parameter θ0.

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